  In this work we shall present a survey on problems and results on singular
holomorphic foliations and Pfaff systems on complex manifolds assuming that
these objects possess invariant analytic varieties. We will focus on recent
results which have been motivated by classical works of Darboux, Poincar\'e and
Painlev\'e on the problem of algebraic integration of singular polynomial
differential equations. We present results on Poincar\'e and Painlev\'e problem
of bounding the degree and the genus of analytic varieties invariant by
holomorphic foliations and Pfaff systems. We shall discuss the general ideas of
the theory of integrability characterizing the existence of meromorphic first
integrals for complex analytic Pfaff equations.
