  Recent studies of unconventional superconductivity have focused on charge or
spin fluctuation, instead of electron-phonon coupling, as an origin of
attractive interaction between electrons. On the other hand, a multipole order,
which represents electrons' degrees of freedom in strongly correlated and
spin-orbit-coupled systems, has recently been attracting much attention.
Stimulated by this background, we investigate multipole-fluctuation-mediated
superconductivity, which proposes a new pairing mechanism of unconventional
superconductivity. Indeed, previous works have shown spin-triplet
superconductivity induced by fluctuations of odd-parity electric multipole
orders in isotropic systems. In this study, we establish a general formulation
of the multipole-fluctuation-mediated superconductivity for all multipole
symmetries, in both isotropic and crystalline systems. As a result, we reveal
various anisotropic pairings induced by odd-parity and/or higher-order
multipole fluctuations, which are beyond the ordinary charge or spin
fluctuations. Topological superconductivity due to the mechanism is also
discussed. Based on the obtained results, we discuss unconventional
superconductivity in doped SrTiO$_3$, PrTi$_2$Al$_{20}$, Li$_2$(Pd, Pt)$_3$B,
and magnetic multipole metals.
