We investigate the dynamics of Brownian particles which are active in the sense that they take up energy from the environment, which can be stored in a internal energy depot and used for different activities. As one example, we consider the generation of a self-consistent field, which in turn affects the movement of the particles. The dynamics can in this case be described by coupled reaction-diffusion equations, but will be more efficiently simulated by means of Langevin equations for the active particles. As another example, we discuss the active motion of Brownian particles which can be described by a non-linear, velocity-dependent friction function. Provided a supercritical supply of energy, the active particles are able to perform non-trivial motion, such as "uphill'' motion against the direction of an external force, or motion on a stochastic limit cycle.