Given a homotopy class of maps between Riemannian manifolds, a classical question in geometric analysis asks whether one can find a representative minimizing some $L^p$ norm of the gradient. Attempts to answer this question lead one to consider Sobolev spaces of manifold-valued maps, where one quickly encounters the problem that homotopy classes may not be well-defined. We will survey a series of results describing the homotopy structure of Sobolev maps, and discuss some related geometric variational problems.