  In the late 1970's, the axion was proposed as a solution to the Strong CP
Problem, i.e. the puzzle why the strong interactions conserve parity P and the
product CP of charge conjugation and parity in spite of the fact that the
Standard Model of elementary particles as a whole violates those symmetries.
The original axion was soon ruled out by laboratory experiments and
astrophysical considerations, but a new version was invented which is much more
weakly coupled and which evades the laboratory and astrophysical constraints.
It was dubbed the invisible" axion. However, the axion cannot be arbitrarily
weakly coupled because it is overproduced in the early universe by vacuum
realignment in the limit of vanishing coupling. The axions produced by vacuum
realignment are a form of cold dark matter today. The axion provides a solution
then not only to the Strong CP Problem but also to the dark matter problem.
Various methods have been proposed to search for dark matter axions and for
axions emitted by the Sun. Their implementation and improvement has led to
significant constraints on the notion of an invisible axion. Even purely
laboratory methods may place significant constraints on invisible axions or
axion-like particles. This review discusses the various methods that have been
proposed and provides theoretical derivations of their signals.
