  This article is an English translation of Japanese article "Musubime to
Kyokumen", Math. Soc. Japan, Sugaku Vol. 67, No. 4 (2015) 403--423. It surveys
a specific area in Knot Theory concerning surfaces in knot exteriors.
  In version 2, we added comments on the solutions or counterexamples for
Conjecture 3.5, Conjecture 3.7 and Conjecture 5.30.
