In the first part of the talk we will review Andreas Floer's construction, from the mid 1980's, of 3-manifold invariants: applying the ideas of Morse Theory to the Chern-Simons functional. We will recall how these Floer homology groups fit into a partial topological field theory and some of the many developments in this area over the past four decades. In the latter part of the talk we will discuss variants of these ideas in higher dimensions and some of the analytical questions involving singularity formation which arise.