This talk concerns the geometric classification of smooth, locally free, codimension-one actions of higher-rank simple Lie groups G (e.g., SL(n,R), n at least 3) on closed manifolds M. We show that M is G-equivariantly diffeomorphic (up to a finite cover) to G/\Gamma x S^1, for a uniform lattice \Gamma of G, if and only if it admits a probability measure that is mixing for a minimal parabolic subgroup P of G. When such a P-mixing probability measure does not exist, we show that M is a fiber bundle over a boundary space of G, i.e., over G/Q for some parabolic subgroup Q of G. This result is in the spirit of the Zimmer program.