  Collapse models explain the absence of quantum superpositions at the
macroscopic scale, while giving practically the same predictions as quantum
mechanics for microscopic systems. The Continuous Spontaneous Localization
(CSL) model is the most refined and studied among collapse models. A well-known
problem of this model, and of similar ones, is the steady and unlimited
increase of the energy induced by the collapse noise. Here we present the
dissipative version of the CSL model, which guarantees a finite energy during
the entire system's evolution, thus making a crucial step toward a realistic
energy-conserving collapse model. This is achieved by introducing a non-linear
stochastic modification of the Schr\"odinger equation, which represents the
action of a dissipative finite-temperature collapse noise. The possibility to
introduce dissipation within collapse models in a consistent way will have
relevant impact on the experimental investigations of the CSL model, and
therefore also on the testability of the quantum superposition principle.
