Discs and other surfaces in 4-manifolds

A popular question in low-dimensional topology is to determine whether a given knot bounds a disc in a given 4-manifold with boundary the 3-sphere. In the first part of the talk I will introduce this problem, discuss its connection with the smooth, 4-dimensional Poincaré conjecture, and describe the state of the art. In the second part I will state and prove some of my results in collaboration with various mathematicians about this problem for some 4-manifolds.