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Colors arise from two main mechanisms: pigments and structures. While pigment-based colors are caused by the absorption of certain wavelengths, structural colors are caused by the reflection of certain wavelengths that are comparable to spatial variations in the refractive index of a material. In this thesis, I study angle-independent structural colors that are formed when the variations are spatially isotropic, and I investigate the physical processes behind these colors with the goal of predicting and designing them for applications. We first identify the physical phenomena behind the spectral features of disordered packings of spherical nanoparticles in a matrix. We confirm that single scattering determines the location of the main color peak, but we also find that multiple scattering contributes to an increase in reflectance toward short wavelengths and that total internal reflection causes a secondary peak. Since these phenomena can alter the color and saturation of a sample, we develop a qualitative design rule based on the transport length, a lengthscale that determines the onset of multiple scattering. The design rule allows us to find the sample thickness required to achieve optimal color saturation. To design specific colors, however, we need an approach to quantitatively predict the effect of multiple scattering on color. We develop a multiple scattering model based on a Monte Carlo technique that simulates the trajectories of photon packets in a film by sampling the scattering directions and step sizes from probability distributions. Following a previously developed single scattering model, we parameterize the Monte Carlo model in terms of the sample properties: the refractive index of the particle and the matrix, the particle radius and volume fraction, and the film thickness. We show that these parameters are not sufficient to achieve quantitative agreement with experimental reflectance spectra. We find that we must account for absorption, even in nearly transparent materials. We also include the effect of particle size polydispersity and a correction factor that accounts for scattering due to roughness at the surface of the film. With these additions, we are able to achieve good agreement with experimental data. When an application has material or structural constraints, we can use the model to compute the gamut of colors that can be achieved or, inversely, to find the parameter values needed to make a specific color. We can also use the model to explore the colors resulting from different configurations, such as core-shell particles or inverse structures. Thus, the model is a powerful tool to understand the effects of physical properties on reflectance spectra and to design angle-independent structural colors for a variety of applications, ranging from paints and coatings to displays and sensors.