[en] The MARS code system, developed over 30 years, is a set of Monte Carlo programs for detailed simulation of hadronic and electromagnetic cascades in an arbitrary geometry of accelerator, detector and spacecraft components with particle energy ranging from a fraction of an electron volt up to 100 TeV. The new MARS15 (2004) version is described with an emphasis on modeling physics processes. This includes an extended list of elementary particles and arbitrary heavy ions, their interaction cross-sections, inclusive and exclusive nuclear event generators, photo - hadron production, correlated ionization energy loss and multiple Coulomb scattering, nuclide production and residual activation, and radiation damage (DPA). In particular, the details of a new model for leading baryon production and implementation of advanced versions of the Cascade-Exciton Model (CEM03), and the Los Alamos version of Quark-Gluon String Model (LAQGSM03) are given. The applications that are motivating these developments, needs for better nuclear data, and future physics improvements are described.