This article provides an overview of quantum memristors, which represent a new direction for neuromorphic applications. The general methodology and concept of constructing a quantum memristor, as well as the possibilities of its implementation on photonic platforms, superconducting systems and ultracold trapped ions, are considered. The latter platform is being actively developed at the Lebedev Physical Institute for conducting quantum computations and building a high-performance quantum computer; it has also proven itself well in the creation of optical clocks.