In this lecture I will present an unconventional perspective on least-squares finite element methods, which connects them to compatible methods and shows that least-squares methods can enjoy the same conservation properties as their mixed Galerkin cousins. To a casual observer, compatible (or mimetic) methods and least squares principles for PDEs couldn't be further apart. Mimetic methods inherit key conservation properties of the PDE, can be related to a naturally occurring optimization problem, and require specially selected, dispersed degrees of freedom. The conventional wisdom about least squares is that they rely on artificial energy principles, are only approximately conservative, but can work with standard C0 nodal (or collocated) degrees of freedom. The latter is considered to be among the chief reasons to use least squares methods. This lecture demonstrates that exactly the opposite is true about least-squares methods. First, I will argue that nodal elements, while admissible in least squares, do not allow them to realize their full potential, should be avoided and are, perhaps, the least important reason to use least squares! Second, I will show that for an important class of problems least squares and compatible methods are close relatives that share a common ancestor, and in some circumstances compute identical answers. The price paid for gaining favorable conservation properties is that one has to give up what is arguably the least important advantage attributed to least squares methods: one can no longer use C0 nodal elements for all variables. If time permits I will explore two other unconventional uses of least-squares ideas which result in numerical schemes with attractive computational properties: a least-squares mesh-tying method that passes patch tests of arbitrary orders, and a locally conservative discontinuous velocity least-squares method for incompressible flows. The material in this talk is drawn from collaborative works with M. Gunzburger (FSU), M Hyman (Tulane), L. Olson (UIUC) and J. Lai (UIUC). Sandia National Laboratories is a multi-program laboratory operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin company, for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-AC04-94AL85000.