Least-squares methods have become increasingly popular for solving a wide variety of partial differential equations (PDEs). Some of the compelling features of this methodology include:

* self-adjoint equations, stemming from the minimization principle;
* good operator 'conditioning', stemming from the use of first-order formulations of the PDE or inverse norms in the minimization principle;
* finite element and multigrid performance that is optimal and uniform in certain parameters (e.g., Reynolds number, Poisson ratio, and wave number), stemming from uniform product norm equivalence results.

This talk will describe basic elements of the least-squares methodology, with illustrations that include equations from fluid dynamics, structural mechanics, and electromagnetics.