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

I will give a quick review of the Sard-Smale theorem and the universal moduli space approach to transversality, discuss the relative merits of classical vs. inhomogeneous perturbations, and sketch proofs of the standard theorems stating that the moduli space of regular J-holomorphic curves in symplectic cobordisms is a smooth orbifold, and that all somewhere injective curves are regular for generic J.