  Two Riemannian manifolds are said to have $C^k$-conjugate geodesic flows if
there exist an $C^k$ diffeomorphism between their unit tangent bundles which
intertwines the geodesic flows. We obtain a number of rigidity results for the
geodesic flows on compact 2-step Riemannian nilmanifolds: For generic 2-step
nilmanifolds the geodesic flow is $C^2$ rigid. For special classes of 2-step
nilmanifolds, we show that the geodesic flow is $C^0$ or $C^2$ rigid. In
particular, there exist continuous families of 2-step nilmanifolds whose
Laplacians are isospectral but whose geodesic flows are not $C^0$ conjugate.
