How many records has lionel messi broken?

These are numerous. The most important are listed below.

1) Most goals in a single season in all competitions (73 goals, 2011-12, all-time record)

2) Most goals in a single Primera Division season (50 goals, 2011-12, all-time record, he surpassed right away Cristiano Ronaldo's 40 goals in 2010-11)

3) Most goals ever in a single Champions League season (14 goals, 2011-12, all-time record, he surpassed Van Nistelrooy & equalised Altafini)

4) Most goals in the history of Spanish Super Cup (10 goals so far, 2009-12, all-time record, he surpassed Raul)

5) Most goals in a single Champions League match (5 goals against Bayer Leverkusen in the round of "16", 2012, all-time record, he surpassed Van Basten, S. Inzaghi, Van Nistelrooy, Shevchenko, Prso & Gomis)

6) Most hat-tricks in a Primera Division season (8 hat-tricks, including two 4's against Valencia & Espanyol, 2011-12, all-time record)

7) Most goals in the history of Club World Cup (4 goals so far, 2009 & 2011, he surpassed Romario, Raul, Anelka, Rooney and Aboutrika)

8) Most ever combined goals & assists in a Primera Division season (50 goals & 16 assists, 2011-12)

9) Most official goals in the history of Barcelona (273 goals so far, 2005-present, he surpassed the Barcelona Legend, Cesar Rodriguez)

10) Most Champions League's Top Scorer awards (4 times in a row, 2009-12, he surpassed Puskas, Eusebio, G. Muller & Papin, all-time record)

11) He broke Pele's (King's of Football) record (75) of most goals in a calendar year (76 goals so far, 2012, he still has time for more 'till December 31)

12) He broke Ronaldo's record (47) of most Barcelona's goals in a single Primera Division season (51 goals, 2010-11, all-time record)

13) Most Best Player, Best Forward, Best Foreign & Best Ibero-American Player awards in Primera Division (4, 4, 3 & 5 times respectively, between 2007-2012, all- time records)

14) Most Best Young Player's awards in Europe/World (4 times in a row, 2005-2008, all-time record)

15) Most Ballon D' Or awards (3 in a row so far, 2009-11, all-time record, shared with Platini, Cruyjiff & Van Basten)

16) Most Best Argentinian Player awards (7 times, 2005, 2007-12 in a row, all-time record, he surpassed the Argentine Legend, Diego Maradona - a 4-time winner)

17) He broke Di Stefano's record (54 two-goal games) with the most 2-goal matches in the history of Primera Division. Thus, Messi scored twice (2 goals in one match) in 55 games in La Liga (November 2012, he needs 22 more two-goal matches to surpass Zarra and become the number one ever in that list too)

and there are lots of less significant ones.