This paper is a survey of very recent results about biholomorphic mappings of the ball in $\Bbb{C}^n$ and in reflexive complex Banach spaces. After recalling existence and regularity results in $\Bbb{C}^n$, we present certain applications including univalence criteria and quasiconformal extension results. We also consider nonuniqueness phenomena for solutions of the Loewner differential equation, and a geometric characterization of Loewner chains which satisfy a growth condition in $t$ based on a generalization of the Carathéodory convergence theorem. Finally we describe some properties of Loewner chains and the Loewner equation on the unit ball of a reflexive complex Banach space.