We present a review on quantum memristors, which define a new direction for neuromorphic applications. We discuss the general methodology and concept of constructing a quantum memristor, as well as the possibility of its implementation on photonic platforms, superconducting systems, and ultracold trapped ions. The latter platform is being actively developed at the Lebedev Physical Institute for conducting quantum computations and for building a high-performance quantum computer; it has also proven its worth in creating optical clocks.