We present a systematic theory of dissipation in finite Fermi systems. This theory is based on the application of periodic-orbit theory to linear response of many-body systems. We concentrate only on the mesoscopic aspect of the phenomena wherein a many-body system can be reduced to a singlebody in an effective mean-field. We obtain semiclassical periodic-orbit corrections on top of the twotime correlation function for the rate of energy dissipation. We show that this energy dissipation is irreversible on an observational time-scale. To do so, we derive a generalised Smoluchowski equation for the energy distribution in the quantal domain. Employing the Weyl-Wigner expansion, we also obtain an equation governing the evolution of energy distribution in the combination of semiclassical and adiabatic approximations. Further, we show how the periodic orbit corrections are related to geometric phase acquired by a single-particle wavefunction as it evolves with the slow, time-varying mean-field. We present our results for the important case of mixed dynamics also. We obtain randommatrix results for response functions. We incorpoate chaos in the underlying classical system by writing an ansatz for a generic wavefunction which leads to various central results of equilibrium statistical mechanics. This new formalism is extended here to include dissipation. Finally, we present an expression for the viscosity tensor encountered in nuclear fission in terms of periodic orbits of single particle in an adiabatically deforming nucleus.