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We consider the application of least-squares variational principles and the finite element method to the numerical solution of boundary value problems arising in the fields of solid and fluid mechanics. For many of these problems least-squares principles offer many theoretical and computational advantages in the implementation of the corresponding finite element model that are not present in the traditional weak form Galerkin finite element model. Most notably, the use of least-squares principles leads to a variational unconstrained minimization problem where stability conditions such as inf-sup conditions (typically arising in mixed methods using weak form Galerkin finite element formulations) never arise. In addition, the least-squares based finite element model always yields a discrete system of equations with a symmetric positive definite coefficient matrix. These attributes, amongst many others highlighted and detailed in this work, allow the development of robust and efficient finite element models for problems of practical importance. The research documented herein encompasses least-squares based formulations for incompressible and compressible viscous fluid flow, the bending of thin and thick plates, and for the analysis of shear-deformable shell structures.