This text is an expanded series of lecture notes based ona 5-hour course given at the workshop entitledWorkshop for youngresearchers: Groups acting on manifoldsheld in Teresópolis, Brazil inJune 2016. The course introduced a number of classical tools in smoothergodic theory—particularly Lyapunov exponents and metric entropy—astools to study rigidity properties of group actions on manifolds.We do not present a comprehensive treatment of group actions or generalrigidity programs. Rather, we focus on two rigidity results in higher-rank dynamics: the measure rigidity theorem for affine Anosov abelianactions on tori due to A. Katok and R. Spatzier and recent the work of theauthor with D. Fisher, S. Hurtado, F. Rodriguez Hertz, and Z. Wang onactions of lattices in higher-rank semisimple Lie groups on manifolds Wegive complete proofs of these results and present sufficient background insmooth ergodic theory needed for the proofs. A unifying theme in this textis the use of metric entropy and its relation to the geometry of conditionalmeasures along foliations as a mechanism to verify invariance of measures.