Disordered solids are all around us from glass and plastic to sand and grains. However, compared to their crystalline counterparts, amorphous materials have unusual properties that are relatively poorly understood. A longstanding question is whether or not the unusual behavior of these systems is structural in origin or whether it is purely dynamical in nature. Here we investigate tools to probe the structure of disordered materials and the role that structure plays in determining dynamics. We begin by investigating a particular class of disordered solids called jammed packings which are composed of finitely repulsive spheres. We show that the stability of these systems with respect to continuous perturbations of their boundary is controlled by a structural length scale, known as the transverse length scale, which diverges as the spheres are decompressed. We then turn to two techniques that are commonly used to measure structural properties of disordered solids in experiment. The first technique, which computes the phonon spectrum from correlations of particle fluctuations, we show can be reliably applied in experiment as long as care is taken that enough data is available for the spectrum to converge. The second technique, which purports to measure local elastic constants, we show to be fundamentally inapplicable to disordered systems, in contrast to the success of the method when applied to crystalline systems. The final sections of the thesis are devoted to examining the role of local structure in determining dynamics in strained glasses and supercooled liquids above the glass transition temperature. We introduce a novel machine learning method for constructing a continuous field, that we call softness, as a coarse graining over local density. We show softness to be more strongly correlated with dynamics than existing methods and demonstrate that it may be applied across a wide range of systems. We leverage the softness picture to show that low temperature glasses under strain can be meaningfully understood in terms of the dynamics of a population of ``soft spots'' in analogy to crystalline systems which are controlled by populations of defects. Finally, we show that the well known heterogeneous dynamics of supercooled liquids arise from a heterogeneous distribution of energy scales in the system, that are in turn correlated with softness. This allows us to construct a simple model for the slow relaxation of glassy liquids that is in excellent agreement with simulation results.