  We study Bogomolny equations on $R^2\times S^1$. Although they do not admit
nontrivial finite-energy solutions, we show that there are interesting
infinite-energy solutions with Higgs field growing logarithmically at infinity.
We call these solutions periodic monopoles. Using Nahm transform, we show that
periodic monopoles are in one-to-one correspondence with solutions of Hitchin
equations on a cylinder with Higgs field growing exponentially at infinity. The
moduli spaces of periodic monopoles belong to a novel class of hyperk\"ahler
manifolds and have applications to quantum gauge theory and string theory. For
example, we show that the moduli space of $k$ periodic monopoles provides the
exact solution of ${\cal N}=2$ super Yang-Mills theory with gauge group $SU(k)$
compactified on a circle of arbitrary radius.
