Tsallis's generalization of statistical mechanics is summarized. A modification of this formalism which employs a normalized expression of the q-expectation value for the computation of equilibrium averages is reviewed for the cases of pure Tsallis statistics and Maxwell-Tsallis statistics. Monte Carlo and Molecular Dynamics algorithms which sample the Tsallis statistical distributions are presented. These methods have been found to be effective in the computation of equilibrium averages and isolation of low lying energy minima for low temperature atomic clusters, spin systems, and biomolecules. A phase space coordinate transformation is proposed which connects the standard Cartesian positions and momenta with a set of positions and momenta which depend on the potential energy. It is shown that pure Tsallis statistical averages in this transformed phase space result in the q-expectation averages of Maxwell-Tsallis statistics. Finally, an alternative novel derivation of the Tsallis statistical distribution is presented. The derivation begins with the classical density matrix, rather than the Gibbs entropy formula, but arrives at the standard distributions of Tsallis statistics. The result suggests a new formulation of imaginary time path integrals wich may lead to an improvement in the simulation of equilibrium quantum statistical averages.