In this chapter, we will discuss some dynamical properties of the Fermi accelerator model in the presence of dissipation. The first type of dissipation considered will be through inelastic collisions, which lead to aFractional energy loss fractional energy loss at each impact of the particle with the walls. We will show that, depending on the choice of control parameters, stableStable manifold and unstableUnstable manifold manifolds of the same saddle-type fixed point can intersect, leading to aBoundary crisis boundary crisis. This crisis destroys the chaotic attractor, replacing it with a chaotic transient. Another type of dissipation to be considered is viscous drag. It is a force that acts along the particle’s trajectory, gradually reducing its velocity. Three types ofDrag force drag forces will be considered: (i) linearly proportional to velocity; (ii) proportional to the square of the velocity; and (iii) proportional to a power of the velocity that is neither linear nor quadratic.