  In a number of field theoretical models the vacuum angle \theta enters
physics in the combination \theta/N, where N stands generically for the number
of colors or flavors, in an apparent contradiction with the expected 2 \pi
periodicity in \theta. We argue that a resolution of this puzzle is related to
the existence of a number of different \theta dependent sectors in a finite
volume formulation, which can not be seen in the naive thermodynamic limit V ->
\infty. It is shown that, when the limit V -> \infty is properly defined,
physics is always 2 \pi periodic in \theta for any integer, and even rational,
values of N, with vacuum doubling at certain values of \theta. We demonstrate
this phenomenon in both the multi-flavor Schwinger model with the bosonization
technique, and four-dimensional gluodynamics with the effective Lagrangian
method. The proposed mechanism works for an arbitrary gauge group.
