This paper describes ZEUS-MP, a multi-physics, massively parallel, message-passing implementation of the ZEUS code. ZEUS-MP differs significantly from the thoroughly documented ZEUS-2D code, the completely undocumented (in peer-reviewed literature) ZEUS-3D code, and a marginally documented “version 1” of ZEUS-MP first distributed publicly in 1999. ZEUS-MP offers an MHD algorithm which is better suited for multidimensional flows than the ZEUS-2D module by virtue of modifications to the Method of Characteristics scheme first suggested by Hawley & Stone (1995). This MHD module is shown to compare quite favorably to the TVD scheme described by Ryu et al. (1998). ZEUS-MP is the first publicly-available ZEUS code to allow the advection of multiple chemical (or nuclear) species. Radiation hydrodynamic simulations are enabled via an implicit flux-limited radiation diffusion (FLD) module. The hydrodynamic, MHD, and FLD modules may be used, singly or in concert, in one, two, or three space dimensions. Additionally, so-called “1.5-D” and “2.5-D” grids, in which the “half-D” denotes a symmetry axis along which a constant but non-zero value of velocity or magnetic field is evolved, are supported. Self gravity may be included either through the assumption of a GM/r potential or a solution of Poisson’s equation using one of three linear solver packages (conjugategradient, multigrid, and FFT) provided for that purpose. Point-mass potentials are also supported. Because ZEUS-MP is designed for large simulations on parallel computing platforms, considerable attention is paid to the parallel performance characteristics of each module in the code. Strong-scaling tests involving pure hydrodynamics (with and without self-gravity), MHD, and RHD are performed in which large problems (256 3 zones) are distributed among as many as 1024 processors of an IBM SP3. Parallel efficiency is a strong function of the amount of communication required between processors in a given algorithm, but all modules are shown to scale well on up to 1024 processors for the chosen fixed problem size. Subject headings: hydrodynamics – methods:numerical – methods:parallel – MHD – radiative transfer