Free probability is a noncommutative analog of probability theory that is tremendously useful for describing the large-N limits of N x N random matrices.  Combining free probability with ideas from stochastic analysis produces a discipline called noncommutative stochastic analysis, which provides a framework for describing the large-N limits of time-dependent N x N random matrices, i.e., N x N matrix stochastic processes.  A foundational example is Philippe Biane's introduction of free Brownian motion, the large-N limit of Brownian motion on the space of N x N Hermitian matrices, in the mid-1990s.  Free Browian motion is an example of a noncommutative stochastic process, but is it an example of a "noncommutative Brownian motion"?  Surprisingly, the latter notion does not presently exist in the literature.  This talk, based on joint work in progress with Collins, Junge, and Speicher, will introduce a proposal for a definition of a noncommutative Brownian motion (NCBM) with classical Brownian motion, Hermitian matrix Brownian motion, free Brownian motion, and more as particular examples.  It will also touch on stochastic calculus for NCBM, which yields the surprising application of a new characterization of classical Brownian motion.