This chapter provides a summary of the equations governing the nonlinear mechanical behavior of magnetoelastic solids that can be subjected to large elastic deformations in the presence of a magnetic field. First of all an overview of the required basic equations of nonlinear elasticity is provided. Next, relevant magnetic field variables and corresponding boundary conditions are summarized in both their Eulerian and Lagrangian forms. The elastic and magnetic equations are then combined to form a theory of nonlinear magnetoelastic interactions. The Lagrangian magnetic vectors are used as the basis for the constitutive equations of a nonlinear magnetoelastic material based on the use of a total energy function, which depends on both mechanical and magnetic variables. In particular, the theory is developed first with the total energy a function of the deformation gradient tensor and the Lagrangian form of the magnetic induction vector as independent variables. An alternative form of the total energy with the magnetic induction vector replaced by the magnetic field vector is also discussed briefly, and this is used in an illustrative application to the deformation of a circular cylindrical tube of magnetoelastic material subject to internal pressure combined with a circumferential magnetic field. Consideration then focuses on the effect of magnetic saturation and remnant magnetization on the constitutive equations, and it is shown how these must take on rather specialized forms. An application of this part of the theory to the equibiaxial deformation of a slab subject to a magnetic field normal to its major faces is then examined. Graphical illustrations are provided for each of the examples.