  Field theories compactified on non-simply connected spaces, which in general
allow to impose twisted boundary conditions, are found to unexpectedly have a
rich phase structure. One of characteristic features of such theories is the
appearance of critical radii, at which some of symmetries are broken/restored.
A phase transition can occur at the classical level, or can be caused by
quantum effects. The spontaneous breakdown of the translational invariance of
compactified spaces is another characteristic feature. As an illustrative
example,the O(N) $\phi^4$ model on $M^3\otimes S^1$ is studied and the novel
phase structure is revealed.
