The fact that a disordered material is not constrained in its properties in the same way as a crystalline one presents significant and yet largely untapped potential for novel material design. However, unlike their crystalline counterparts, disordered solids are not well understood. Though currently the focus of intense research, 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 a highly idealized system composed of frictionless soft spheres at zero temperature that, when compressed, undergos 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 that is sorely needed to develop 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 a new class of contact nonlinearities," demonstrating that the leading order behavior of such solids can be ascertained by linear response. We then investigate the critical nature of the jamming transition, focusing on two diverging length scales and the importance of finite-size effects. Next, we argue that this jamming transition plays the same role for disordered solids as the idealized perfect crystal plays for crystalline solids. Not only can it be considered an idealized starting point for understanding the properties of disordered materials, but it can even influence systems that have a relatively high amount of crystalline order. As a result, the behavior of solids can be thought of as existing on a spectrum, with the perfect crystal at one end and the jamming transition at the other. Finally, we introduce a new principle for disordered solids 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 of one another and provides a great deal of flexibility in designing materials with unique, textured and tunable properties.