  Stochastic methods are ubiquitous to a variety of fields, ranging from
Physics to Economy and Mathematics. In many cases, in the investigation of
natural processes, stochasticity arises every time one considers the dynamics
of a system in contact with a somehow bigger system, an environment, that is
considered in thermal equilibrium. Any small fluctuation of the environment has
some random effect on the system. In Physics, stochastic methods have been
applied to the investigation of phase transitions, thermal and electrical
noise, thermal relaxation, quantum information, Brownian motion etc.
  In this review, we will focus on the so-called stochastic Schr\"odinger
equation. This is useful as a starting point to investigate the dynamics of
open quantum systems capable of exchanging energy and momentum with an external
environment. We discuss in some details the general derivation of a stochastic
Schr\"odinger equation and some of its recent applications to spin thermal
transport, thermal relaxation, and Bose-Einstein condensation. We thoroughly
discuss the advantages of this formalism with respect to the more common
approach in terms of the reduced density matrix. The applications discussed
here constitute only a few examples of a much wider range of applicability.
