In this paper we address the problem of existence of inverse integrating factors for an analytic planar vector field in a neighborhood of its nonwandering sets. It is proved that there always exists a smooth inverse integrating factor in a neighborhood of a limit cycle, obtaining a necessary and sufficient condition for the existence of an analytic one. This condition is expressed in terms of the Ecalle–Voronin modulus of the associated Poincare map. The existence of inverse integrating factors in a neighborhood of an elementary singularity is also established, and we give the first known examples of analytic vector fields in R not admitting a C inverse integrating factor in any neighborhood of either a limit cycle or a weak focus. Moreover, it is shown that a C inverse integrating factor of a C planar vector field must vanish identically on the polycycles which are limit sets of its flow, thereby solving a problem posed by Garcia and Shafer (J. Differential Equations 217 (2005) 363–376).