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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.