  We present some transversality results for a category of Fr\'{e}chet
manifolds, the so-called $MC^k$-Fr\'{e}chet manifolds. In this context, we
apply the obtained transversality results to construct the degree of nonlinear
Fredholm mappings by virtue of which we prove a rank theorem, an invariance of
domain theorem and a Bursuk-Ulam type theorem.
