Most basic differential topology theorems carry over to the equivariant case with mild modifications; see for instance Wasserman's paper. One thing that fails (more or less obviously) is equivariant transversality - the only $\Bbb Z/2$-equivariant map $\Bbb R^n \to \Bbb R^n$, the first equipped with the trivial action and the second, negation, is constant, so obviously we cannot equivariantly perturb this to be transverse to something or other.
Of course, it's still desirable to find conditions for which we can achieve equivariant transversality; or appropriate definitions so that we get something manifoldish when we cut something out equivariant-transversely. I can find a few different notes on the subject (eg 1 2 3) but I'm hoping to find something authoritative and reasonably up to date about what's known.
Is there some modern, reasonably comprehensive survey of the definitions and results on equivariant transversality? I am interested in the compact Lie case but even the case of a specific finite group would be great.
