"Pawns are born free, but everywhere are in chains" -- Andrew SOLTIS

1. A pawn chain is a diagonal line of pawns where each pawn is protected by one to the rear.
2. There is a very common example in the main line of the French Defence: 1. e4 e6 2. d4 d5 3. e5 c5 4. c3 Nc6 5. Nf3 Qb6
3. 

   

   

   

   

   

   

   
   White has a pawn chain from b2 to e5
4. Nimzowitsch taught that the pawn chain should be attacked at its base. So Black should attack c3 with ...b5-b4, and White should attack e6 with f4-f5.
5. Black can swap on d4: ...cxd4, cxd4: which brings the base of the pawn chain to d4, which is easier to attack, and easier to defend.
6. But these days Black is often content to attack e5 with ...f5, instead or as well.
7. A common continuation is 6.a3 c4, extending the Black pawn chain one more step.
8. 

   

   

   

   

   

   

   
9. I enclose some games with spectacular pawn chains

Click [...] to see list of games

[Event "?"]
[Site "?"]
[Date "????.??.??"]
[Round "?"]
[White "Enter New Game"]
[Black "?"]
[Result "*"]
[PlyCount "12"]

1. e4 e6 2. d4 d5 3. e5 c5 4. c3 Nc6 5. Nf3 Qb6 6. a3 c4 *


[Event "Miskolc Avas op"]
[Site "Miskolc"]
[Date "1999.??.??"]
[Round "7"]
[White "Cselenyi, Balazs"]
[Black "Szabo, Janos"]
[Result "1-0"]
[ECO "E99"]
[WhiteElo "2045"]
[BlackElo "2000"]
[PlyCount "71"]
[EventDate "1999.??.??"]
[EventType "swiss"]
[EventRounds "9"]
[EventCountry "HUN"]
[SourceTitle "EXT 2000"]
[Source "ChessBase"]
[SourceDate "1999.11.16"]
[SourceVersion "1"]
[SourceVersionDate "1999.11.16"]
[SourceQuality "1"]

1. d4 Nf6 2. c4 g6 3. Nc3 Bg7 4. e4 d6 5. Nf3 O-O 6. Be2 e5 7. O-O Nc6 8. d5 Ne7 9. Ne1 Nd7 10. Be3 f5 11. f3 Nf6 12. b4 f4 13. Bf2 g5 14. c5 Ng6 15. Nd3 Bd7 16. Qb3 Kh8 17. b5 b6 18. cxd6 cxd6 19. Nb4 Ne8 20. Nc6 Bxc6 21. bxc6 Nc7 22. a4 Qe7 23. Nb5 Rfb8 24. Qa3 Bf8 25. Nxc7 Qxc7 26. a5 b5 27. a6 Be7 28. Rfb1 h5 29. Rxb5 Rxb5 30. Bxb5 Bd8 31. Be2 Rb8 32. Qa2 Ne7 33. Rb1 Nc8 34. Rb7 Rxb7 35. axb7 Nb6 36. Qxa7 1-0

[Event "WMSG Blitz Pair 1st"]
[Site "Beijing"]
[Date "2008.10.08"]
[Round "9"]
[White "Vasilevich, Tatjana"]
[Black "Socko, Monika"]
[Result "1-0"]
[ECO "E11"]
[WhiteElo "2368"]
[BlackElo "2434"]
[PlyCount "115"]
[EventDate "2008.10.08"]
[EventType "swiss (blitz)"]
[EventRounds "11"]
[EventCountry "CHN"]
[SourceTitle "CBM 126 Extra"]
[Source "ChessBase"]
[SourceDate "2008.11.05"]
[SourceVersion "1"]
[SourceVersionDate "2008.11.05"]
[SourceQuality "1"]

1. d4 Nf6 2. c4 e6 3. g3 Bb4+ 4. Bd2 a5 5. Bg2 O-O 6. Nf3 d6 7. O-O Nc6 8. Bg5 a4 9. a3 Ba5 10. Qxa4 Bd7 11. Qc2 h6 12. Bf4 d5 13. c5 Nh5 14. Ra2 e5 15. Bxe5 f6 16. Bf4 Nxf4 17. gxf4 Bg4 18. e3 Qe7 19. Qb3 Rfd8 20. Qxb7 Qe6 21. Qb3 Ne7 22. Qc2 Bf5 23. Qe2 c6 24. b4 Bc7 25. Nc3 Ng6 26. a4 Bg4 27. Rfa1 Nh4 28. b5 Qf7 29. h3 Bh5 30. b6 Qg6 31. Nxh4 Qf7 32. Bf3 Bg6 33. Nxg6 Qxg6+ 34. Bg2 Bb8 35. a5 Re8 36. Qf3 Ra6 37. Qe2 Ra8 38. Qf1 f5 39. a6 Re7 40. a7 Bxf4 41. exf4 Qf6 42. Qd3 g6 43. Rd2 Kh7 44. Re2 Rf7 45. Re5 g5 46. fxg5 Qxg5 47. Qe3 Qg6 48. Kh2 f4 49. Qf3 Rg7 50. Rae1 Rag8 51. Rg1 Qc2 52. Ne2 Qd2 53. Nxf4 Qxd4 54. Qd3+ Qxd3 55. Nxd3 d4 56. Nf4 Rf8 57. Ne6 d3 58. Nxf8+ 1-0

[Event "Berlin"]
[Site "Berlin"]
[Date "1935.??.??"]
[Round "5"]
[White "Gerhardt, Walter"]
[Black "Ernst, Wilhelm"]
[Result "1/2-1/2"]
[ECO "D34"]
[PlyCount "90"]
[EventDate "1935.??.??"]
[EventType "tourn"]
[EventRounds "9"]
[EventCountry "GER"]
[SourceTitle "EXT 2008"]
[Source "ChessBase"]
[SourceDate "2007.11.25"]
[SourceVersion "1"]
[SourceVersionDate "2007.11.25"]
[SourceQuality "1"]

1. d4 d5 2. c4 e6 3. Nc3 c5 4. cxd5 exd5 5. Nf3 Nc6 6. g3 Nf6 7. Bg2 Be7 8. O-O O-O 9. Bg5 Be6 10. Rc1 c4 11. Ne5 Qb6 12. Nxc6 bxc6 13. e4 h6 14. Bxf6 Bxf6 15. e5 Be7 16. b3 Ba3 17. Na4 Qa5 18. Rc3 Bb4 19. Rc1 Bf5 20. bxc4 Bd2 21. cxd5 Bxc1 22. dxc6 Ba3 23. d5 Rab8 24. Qf3 Bc2 25. Qxa3 Qxa4 26. Qxa4 Bxa4 27. c7 Rb2 28. Rc1 Bd7 29. d6 Rc8 30. Bd5 Kf8 31. Re1 Rb6 32. f4 Ke8 33. Kf2 Be6 34. Be4 g6 35. Ke3 Kd7 36. a4 Bf5 37. Bd5 Be6 38. Be4 Bf5 39. Bd5 Be6 40. Bg2 Rb3+ 41. Kd4 Rb2 42. Bh1 Rd2+ 43. Ke3 Rxd6 44. exd6 Kxd6 45. Rc1 Rxc7 1/2-1/2

[Event "URU-ch"]
[Site "Uruguay"]
[Date "1988.??.??"]
[Round "?"]
[White "Candan, Gonzalo"]
[Black "Izquierdo, Daniel"]
[Result "0-1"]
[ECO "E97"]
[BlackElo "2250"]
[PlyCount "84"]
[EventDate "1988.??.??"]
[EventType "tourn"]
[EventRounds "15"]
[EventCountry "URU"]
[SourceTitle "EXT 2001"]
[Source "ChessBase"]
[SourceDate "2000.11.22"]
[SourceVersion "1"]
[SourceVersionDate "2000.11.22"]
[SourceQuality "1"]

1. d4 Nf6 2. c4 g6 3. Nc3 Bg7 4. e4 d6 5. Nf3 O-O 6. Be2 e5 7. O-O Nc6 8. d5 Ne7 9. b4 a5 10. Ba3 axb4 11. Bxb4 Nd7 12. Nd2 f5 13. f3 h5 14. Nb3 Bh6 15. Kh1 Be3 16. a4 f4 17. a5 g5 18. Na4 g4 19. Qe1 Ng6 20. c5 Qg5 21. Bd2 Bxd2 22. Qxd2 Qh4 23. Bd1 g3 24. h3 Nf6 25. Kg1 Bxh3 26. gxh3 Qxh3 27. Qg2 Qc8 28. Qh1 h4 29. Kg2 Nh7 30. Qh3 Ng5 31. Qxc8 Raxc8 32. Re1 h3+ 33. Kh1 Kg7 34. Nc3 Nh4 35. Nd2 Rh8 36. Be2 h2 37. Red1 Nh3 38. Rf1 Kf6 39. c6 Rcg8 40. Nd1 g2+ 41. Kxh2 g1=Q+ 42. Rxg1 Nxf3+ 0-1