This is a modern textbook for courses in continuum mechanics. It provides both the theoretical framework and the numerical methods required to model the behavior of continuous materials. This self-contained textbook is tailored for advanced undergraduate or first year graduate students with numerous step-by-step derivations and worked out examples. The author presents both the general continuum theory and the mathematics needed to apply it in practice. The derivation of constitutive models for ideal gasses, fluids, solids, and biological materials and the numerical methods required to solve the resulting differential equations are also detailed. Specifically, the text presents the theory and numerical implementation for the finite difference and the finite element methods in the Matlab® programming language. It includes thirteen detailed Matlab® programs illustrating how constitutive models are used in practice.