Recent advances in molecular dynamics methodology have made it possible to study routinely the microscopic details of chemical processes in the condensed phase using high-speed computers. Thus, it is timely and useful to provide a pedagogical treatment of the theoretical and numerical aspects of modern molecular dynamics simulation techniques and to show several applications that illustrate the capability of these approaches. First, the standard Newtonian or Hamiltonian dynamics based method is presented followed by a discussion of theoretical advances related to non-Hamiltonian molecular dynamics. Examples of non-Hamiltonian molecular dynamics schemes capable of generating the canonical and isothermal-isobaric ensemble are analyzed. Next, the novel Liouville operator factorization approach to numerical integration is reviewed. The power and utility of this new technique are contrasted to more basic methods, particularly, in the development of multiple time scale and non-Hamiltonian integrators. Since the results of molecular dynamics simulations depend on the interparticle interactions employed in the calculations, modern empirical force fields and ab initio molecular dynamics approaches are discussed. An example calculation combining an empirical force field and novel molecular dynamics methods, the mutant T4 lysozyme M61 in water, will be presented. The combination of electronic structure with classical dynamics, the so called ab initio molecular dynamics method, will be described and an application to the structure of liquid ammonia discussed. Last, it will then be shown how the classical molecular dynamics methods can be adapted for quantum calculations using the Feynman path integral formulation of statistical mechanics. An application, employing both path integrals and ab initio molecular dynamics, to an excess proton in water will be presented.