$p$-Adic and Adelic Quantum Mechanics

$p$-Adic mathematical physics emerged as a result of efforts to find a non-Archimedean approach to the space–time and string dynamics at the Planck scale. One of its main achievements is a successful formulation and development of $p$-adic and adelic quantum mechanics, which have complex-valued wave functions of $p$-adic and adelic arguments, respectively. Various aspects of these quantum mechanics are reviewed here. In particular, the corresponding Feynman's path integrals, some minisuperspace cosmological models, and relevant approaches to string theory are presented. As a result of an adelic approach, $p$-adic effects exhibit a space–time and some other discreteness, which depend on the adelic quantum state of the physical system under consideration. In addition to the review, this article also contains some new results.