this talk concerns singular values of m-fold products of i.i.d. right-unitarily invariant n x n random matrix ensembles. as n tends to infinity, the height function of the lyapunov exponents converges to a deterministic limit by work of voiculescu and nica-speicher for m fixed and by work of newman and isopi-newman for m tending to infinity with n. in this talk, i will show for a variety of ensembles that fluctuations of these height functions about their mean converge to explicit gaussian fields which are log-correlated for m fixed and have a white noise component for m tending to infinity with n. these ensembles include rectangular ginibre matrices, truncated haar-random unitary matrices, and right-unitarily invariant matrices with fixed singular values. i will sketch our technique, which derives a central limit theorem for global fluctuations via certain conditions on the multivariate bessel generating function, a laplace-transform-like object associated to the spectral measures of these matrix products. this is joint work with vadim gorin.