It is a basic fact that the space of continuous functions on the interval [0,1] contains an isometric copy of every separable metric space, and in particular of every finite metric space. However, it is a subtle question to decide whether or not a given metric space is universal in the sense that it contains a copy of every finite metric space with O(1) bi-Lipschitz distortion. Such questions are difficult open problems even for concrete classical spaces, such as the space of probability measures on the Euclidean plane, equipped with the transportation cost metric. Quantitative versions of the universality problem relate to the notion of “metric cotype,” which is an outgrowth of the Ribe program that serves as a useful invariant with several applications in metric geometry. In a related direction, among the consequences of the Ribe program is the fact that certain sequences of metric spaces exhibit a dichotomic behavior when one tries to embed them into a given metric space: they either embed with constant distortion, or their distortion must grow rapidly. In this talk we will explain these issues while highlighting progress that was obtained in recent years, subtle examples, and several challenging open questions.