Non-trivial space-time topology leads to the possibility of twisted fields viewed as cross sections of non-product vector bundles. For globally hyperbolic space-times twisted real and complex scalar fields are especially interesting, and are in one-to-one correspondence with certain groups determined by the space-time topology. Twisted fields can be quantized and lead to results differing from the usual ones. For example, spontaneous symmetry breaking may be suppressed and regularized vacuum self-energies take on different values. Sets of twisted fields may be collected together into a type of super-multiplet whose size is determined by the space-time topology.