The fact that a disordered material is not constrained in its properties in the same way as a crystal presents significant and yet largely untapped potential for novel material design. However, unlike their crystalline counterparts, disordered solids are not well understood. One of the primary obstacles is the lack of a theoretical framework for thinking about disorder and its relation to mechanical properties. To this end, we study an idealized system of frictionless athermal soft spheres that, when compressed, undergoes a jamming phase transition with diverging length scales and clean power-law signatures. This critical point is the cornerstone of a much larger "jamming scenario" that has the potential to provide the essential theoretical foundation necessary for a unified understanding of the mechanics of disordered solids. We begin by showing that jammed sphere packings have a valid linear regime despite the presence of "contact nonlinearities." We then investigate the critical nature of the transition, focusing on diverging length scales and finite-size effects. Next, we argue that jamming plays the same role for disordered solids as the perfect crystal plays for crystalline solids. Not only can it be considered an idealized starting point for understanding disordered materials, but it can even influence systems that have a relatively high amount of crystalline order. The behavior of solids can thus be thought of as existing on a spectrum, with the perfect crystal and the jamming transition at opposing ends. Finally, we introduce a new principle wherein the contribution of an individual bond to one global property is independent of its contribution to another. This principle allows the different global responses of a disordered system to be manipulated independently and provides a great deal of flexibility in designing materials with unique, textured and tunable properties.