Appears in collection : Summer School 2015: Moduli Problems in Symplectic Geometry

Moduli spaces of pseudoholomorphic curves arise as the zero set of a Fredholm section of a suitable bundle, and one expects and hopes that they can be regularized in order to define invariants that are stable under perturbations. This lecture provides an overview of some of the analytic difficulties that must be solved in order to construct such a regularization, and briefly explains some traditional approaches to their solution, namely via geometric regularizations and finite dimensional reductions.