Abstract: Many classes contain a universal object. For instance, every compact metrizable space is a continuous image of the Cantor set (the Cantor set is projectively universal), and embeds isomorphically into the Hilbert cube (the Hilbert cube is injectively universal). Any separable Banach space is a quotient of $\ell_1$, and embeds isometrically into $C[0,1]$. We discuss the following topics: (1) The existence of (injectively) universal objects that are (almost) homogeneous - that is, any isometry between two finite subsets of such an object extends (or almost extends) to an isometry of the object itself. (2) The existence of universal objects for specific classes of Banach spaces (such as reflexive spaces, or spaces with a separable dual). (3) Universal objects for Banach lattices (based on the recent work with M.-A. Gramcko-Tursi and others).