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Shape-programmed shells morph from flat into curved shapes upon stimulation by light, heat, or chemistry. They are ubiquitous throughout biology, and their synthetic counterparts show great promise as soft large-strain actuators. I’ll present several theoretical/computational advances towards assembling a zoo of mechanically strong shape-morphing "mechanisms". A central theme is encoding Gauss curvature (GC) via patterns of in-plane deformation, due to the mechanical strength inherited by the resultant structures, as Gauss understood centuries ago. The canonical example is a pattern of azimuthal contraction that morphs a planar disk into a cone, which cannot be flattened without energetically costly stretch because its tip bears concentrated GC. In that spirit, I’ll demonstrate novel designs for nematic patterns that encode concentrated GC at generalized "tips" (via topological defects), along ridges (via seams between smooth patterns), and within the central holes of annuli (via spirals). Then, to investigate mechanical strength more quantitatively, we’ll turn to the load-bearing capacity of perfect conical shells. This classical-sounding problem is in fact rather subtle; I will present a new boundary-layer solution, leading to an asymptotic critical force $propto t^{5/2}$. This surprising scaling is novel, and has broad implications for shell buckling more generally. I also explore deep postbuckling, finding further instabilities producing intricate states with multiple Pogorelov-type curved ridges arranged in concentric circles or Archimedean spirals. Finally, I investigate the forces exerted by such states, which limit lifting performance in shape-morphing cones.