The Scot has enjoyed the best year of his career and starting brightly, breaking Federer in the early stages.
Federer responded brilliantly to take a tense first set tie-break before rattling off the second in convincing fashion. The Swiss star ended Murray's tournament with a supreme display to advance to tomorrow's final, where he will play Novak Djokovic.
Read how a frustrating encounter unfolded for Murray with Sports Mole's minute by minute updates below.
7.38pmEvening tennis fans, thanks for joining us at Sports Mole for what promises to a titanic battle between two of the sport's top players.
7.40pmAndy Murray is aiming to cap off a breakthrough year with victory at the 02 Arena while Roger Federer will call on all his experience to prove he is still the man to beat.
7.41pmFederer had a 12-match winning steak in this tournament ended by Juan Martin del Potro last night but remains a formidable opponent.
7.43pmMurray recovered well from defeat to Novak Djokovic in his opening group match, rebounding to beat Jo-Wilfried Tsonga and Thomas Berdych and qualify for the final four.
7.44pmThe Scot just wrapped up his pre-match interview. He said he needs to be aggresive from the outset to claim victory.
7.45pmFederer insists that he feels good despite losing last night. Three punishing sets against Del Potro may prove a factor in tonight's match.
7.47pmBoth players are making their way out on court at an impressive 02 Arena which has really embraced this end of season event.
7.48pmMurray gets a fantastic reception from the London crowd, which is another sell-out.
7.49pmFederer also receives a warm reception, this has all the makings of an awesome match.
7.50pmMurray is aiming to cap a year of remarkable achievement by becoming the first British player to win this tournament since 1970. No pressure then Andy.
7.51pmThe formalities of warming up are underway and we are just minutes away from kicking things off at the 02 Arena. A reminder that the winner will face Novak Djokovic in tomorrow's final.
7.52pmDjokovic struggled early on against Del Potro this afternoon but ground out a vital win against the big-serving Argentine. He will be an interested onlooker for this one.
7.54pmLooking through the stats, Murray is one of very few players to have an advantage over the Swiss star on their head-to-head records. He leads Federer 10-8 in matches played.
7.56amWe're about to get underway, Federer is first to serve.
7.57amMurray wins the opening point and is quickly 0-30 ahead after a rare unforced error from Federer.
Two Break PointsMurray gets the better of the first baseline exchange to earn two break points.
Game MurrayFederer gets aggresive at the net to save the first break point but Murray responds with a huge return to take the early break.
8.01pmTime for Murray to try and consolidate with his opening service game.
8.02pmAn awesome rally from the Scot, mixing things up with a cross-court backhand winner.
8.03pmEarly signs of a wobble on the Murray serve, a cheap error puts Fed 30-15 ahead.
8.04pmMurray fights back to 30-30 and again gets the better of a long rally to earn game point.
Murray 2-0 FedererFederer's attempt at a forehand winner drifts wide to give Murray the game. 2-0 early in the first set to the US Open champion.
8.07pmFederer is back on serve and needs to stamp his authority on this match after a shaky start. Murray's movement forces him into a pair of errors and he's down 15-30.
DeuceFederer recovers well to win the next two points but another exceptional Murray return levels things up at Deuce.
Murray 2-1 FedererMurray cracks a huge forehand a few inches wide to go game-point down. Federer sends a strong serve wide of the Scot to earn his first game of the match.
8.12pmMurray unleashes a 131mph serve to start his second service game and quickly establishes a 30-0 lead.
8.13pmA first unforced error briefly interrupts Murray's flow but Federer can't take advantage, missing an easy forehand at the net.
Game MurrayAndy recovers his composure and holds to restore his two-game advantage. Murray 3-1 Federer
8.16pmFederer is back on serve and looks to be building some momentum. He follows up his fastest serve of the match with a massive forehand to go 40-30 ahead.
Game FedererA clinical volley at the net keeps Federer alive in the first set. He remains a break down but looks to be far more assured after a shaky start.
8.20pmMurray smashes a 132mph serve down the baseline but makes an unforced error on his forehand in the very next point.
8.21pmPressure on the Murray serve for the first time as Federer gets aggresive at the net to make the score 15-30.
8.22pmStrong response from Murray, smacking a huge first serve out of reach.
8.23pmDeja vu in the next point, a well-placed serve earns the Scot game point.
DeuceFederer forces Murray wide with a backhand return and he can't keep the ball in. Level at deuce in an important game in the first set.
8.24pmA superb response from Murray, punishing a short lob with a beautiful backhand slice.
Game MurrayThe Scot repeats the trick at the net to earn a vital hold under severe pressure. He leads 4-2 in the first set.
8.26pmFederer looks in far better shape on serve after a poor start. He uses pace and variety to edge 30-15 ahead.
Game FedererMurray stays aggresive on return but makes a pair of unforced errors to hand Federer a comfortable hold. Murray 4-3 Federer
8.30pmThe eighth game sees new balls introduced but Murray struggles to take advantage, quickly falling 0-30 behind on serve.
8.31pmFederer lets Murray off the hook with a pair of short forehands to level the game at 30-30.
Break PointThe Swiss star gets aggresive at the net and it pays dividends. Big chance for Federer.
Game FedererA magnificent forehand from Federer levels the first set at four games all.
8.34pmFederer is really punishing a slight drop in intensity from the world number three. He rattles off a quick hold to go 5-4 ahead.
Murray 4-5 FedererAndy is serving to stay alive in the first set, wow what a quick turnaround. He starts well and builds a 30-15 lead.
8.39pmFederer fights back and we are all square at 30-30 in this decisive game. Murray then gets aggresive and the plan works with a pair of overhand smashes to earn game point.
Game MurrayAn important hold and we are level in a topsy-turvy opening set. All to play for heading into the 11th game. Murray 5-5 Federer
8.43pmHard to separate these two at this stage as Federer makes a positive start on serve to lead 30-15.
8.44pmFederer earns the hold in confident fashion with his second ace of the match. He leads 6-5 and Murray will once again have to serve to stay in the first set.
Federer 6-5 MurrayPressure on for the Scot and he starts strongly in a crucial game. Federer has a clear tactic to try and go for winners earlier than usual, perhaps to compensate for tired legs after playing last night.
8.49pmA special shot from Federer, the best of the match so far. He finds a backhand winner from an impossible angle of the baseline, incredible stuff.
Tie-break time!Murray holds on to force the first set to a deciding tie-break.
1-0 FedererRoger starts well with a massive forehand volley at the net to edge ahead.
2-1 MurrayThe Scot repeats the trick to move ahead.
3-1 MurrayFederer lashes a backhand wide for the first mini-break, handing Murray a two-point lead.
3-2 MurrayThe world number two quickly responds to reduce the deficit.
3-3Simply awesome from Federer as he wins the longest rally of the match with an unstoppable forehand drive.
4-3 MurrayA huge first serve from Murray makes the difference and gives him a one-point advantage.
4-4Nothing to choose between the players once again as Federer responds with a first serve of his own.
5-4 FedererThe Swiss star seizes the initiative to go infront.
6-4 Set Point FedererTwo sets points for Federer and Murray smashes his racket in anger after coming up short in a baseline battle.
6-5 FedererMurray makes Federer pay for a short return to stay alive in the first set, barely.
Set FedererAnother missed forehand from Murray gifts the Federer the first set after nearly an hour of play. Incredible stuff and a fine response from Fed after a sluggish start.
9.01pmMurray starts on serve in the second set and needs to regain concentration quickly. Federer is flying and takes the opening point with a sharp backhand.
9.03pmAn awesome ace helps Murray rattle off the next four points to earn an important hold.
9.06pmFederer continues his impressive from with a hold to love. Federer 7-6 1-1
9.07pmMurray uses his lightening speed to reach a Federer drop shot before unleashing an emphatic rally to win his first point back on serve.
9.08pmFederer fails to punish a poor drop shot from Murray this time and falls 40-0 love down on the Scot's serve.
9.10pmAnother drop shot from Murray but this time he is made to pay. 40-30
DeuceFederer fights back to stop Murray taking what seemed like a routine hold. Tough to take for the Scot but he needs to bounce back quickly.
Break PointSimply awesome stuff from Federer to take the early break in the second set. Its a long way back now for Murray.
Murray 6-7 1-2Murray looked to be cruising to a much-needed service game but suddenly dropped away and was brutally punished.
9.15pmMurray loses a challenge to start the next game but recovers well to draw level at 15-15.
9.16pmFederer's forehand has been far and away the most potent weapon in this match. More superb stuff puts Federer 40-15 ahead.
Game FedererMurray continues to swim against the tide as Federer consolidates his break to take a 3-1 lead early in the second set.
9.20pmVery important for Murray to re-establish a strong serve and he responds well to go 40-0 infront.
Game MurrayThe Scot holds to love to stay in contention, much more like it for Murray fans. Murray 7-6 2-3 Froch
9.22pmTime for Murray to try and put some pressure on Federer's serve if he wants to keep his season alive.
9.23pmFederer slams the door on Murray's aggresive approach to storm into a 40-0 lead.
Game FedererThe Swiss star restores his two-game advantage with more imperious serving under pressure.
9.25pmMurray is back on serve and back under pressure. He's 15-30 down in a must-win game.
Break PointNice and aggresive from Federer on the second serve to earn a big break point.
Game FedererMurray is punished on second serve by s gorgeous forehand from Federer who is within touching distance of a straight-sets win.
9.29pmFederer has been unplayable at times tonight and races into a 30-0 lead on serve for the match.
9.30pmMurray holds back the tide with a forehand winner but only for a brief moment. Two match points to Federer.
Game Set and Match!Federer wraps things up with minimal fuss. A huge statement from the 17-time Grand Slam champion that he is far from finished at the top level.
9.31pmA rather tepid end to a wonderful season for Murray, who races out of the 02 in double-quick time.
9.32pmFederer responded brilliantly to come back form an early back to dominate from the first-set breaker. He advances to face Novak Djokovic in tomorrow's final.
9.33pmThe Swiss star was supreme and he thanks "an electric" crowd in the post-match interview. He is a six-time winner of the World Tour Finals and clearly enjoys the 02 Arena.
9.35pmFederer: "I keep practicing hard infront of no one for nights like this, it inspires me. It's been an amazing year so to get a chance to play Novak is really nice. I gave everything I had and I can't wait for tomorrow."
9.36pmThanks for joining us here at Sports Mole on a superb day of semi-final action in London. Please come back tomorrow for what promises to be an epic final between Federer and Djokovic.