PythonMHD is a new software package for astrophysical magnetohydrodynamic (MHD) simulations. Although it is a widely understood programming language in the physical sciences, Python has never previously been used to develop a comprehensive, research-oriented MHD simulation code. All of the existing MHD simulation codes are written in lower-level languages, such as C, C++, and FORTRAN. These programming languages are difficult to interpret and, thereby, exacerbate the learning curves associated with MHD software packages. The existing simulation codes further complicate the user’s experience by requiring separate software for data visualization and analysis. PythonMHD provides built-in tools for visualizing and analyzing simulation data while a simulation is still in progress, allowing the user to continuously monitor the evolution of their simulated system. In order to further reduce the likelihood of wasting the user’s time and computational resources on unproductive simulations, PythonMHD performs automatic error checking to assess whether the user’s simulation parameters and initial conditions are likely to generate numerical instabilities. This thesis describes the algorithms that are implemented in PythonMHD and documents their performance on standard 1D, 2D, and 3D MHD test problems. After using these test problems to demonstrate the accuracy of PythonMHD, this document presents a PythonMHD interstellar medium (ISM) turbulence generator, which is currently being used for novel research on astrophysical plasma lensing. In addition to the research applications of PythonMHD, this document explores the educational applications of PythonMHD by discussing how it has served as a teaching tool in a fourth year computational physics course (PHYS 4250) at the University of Manitoba.