In this chapter, we introduce and explore three families of rotationally invariant distributions on \(\mathbb {R}^d\) —the beta, beta prime, and Gaussian distributions—which will play a central role in the chapters that follow. These three families are collectively referred to as beta-type distributions. We then discuss two key properties of beta-type distributions: invariance under projections and invariance under slicing. As applications to problems in geometric probability, we derive explicit formulas for the volumes of spherical caps, the distributions of interpoint distances, as well as certain angle distributions. For the latter, we also describe their high-dimensional limits.