The paper is devoted to the study of integrability problems for vector fields on two dimensional manifolds. The concepts of weak (local) and strong (global) first integrals are discussed. It is proved that every local flow
on a two-dimensional manifold
always possesses a continuous first integral on each component of
is the set of all separatrices of
. Moreover, the existence and uniqueness of an analytic inverse integrating factor in a neighborhood of a strong focus, of a nonresonant hyperbolic node and of a Siegel hyperbolic saddle is also proved.