There are easy examples showing that classical transversality methods cannot always succeed for multiply covered holomorphic curves, but the situation is not hopeless. In this talk I will describe two approaches that sometimes lead to interesting results: (1) analytic perturbation theory, and (2) splitting the normal Cauchy-Riemann operator of a curve along irreducible representations of its automorphism group. Both were pioneered by Taubes in his work on the Gromov invariant and Seiberg-Witten theory in the 1990's, and I will illustrate them by sketching two proofs that the multiply covered holomorphic tori counted by the Gromov invariant are regular for generic J. If time permits, I will discuss some ideas as to how both methods can be applied more generally.