We study projection-free methods for constrained geodesically convex and nonconvex optimization on Riemannian manifolds. In particular, we propose a Riemannian version of the Frank-Wolfe (RFW) method. In the geodesically constraint setting, we show global, non-asymptotic subli- near convergence. We also present a setting under which RFW can attain a linear rate. We further introduce a stochastic RFW for nonconvex optimization. In addition we consider two variance-reduced approaches for finite-sum settings. We then specialize RFW to the PSD manifold and show, that in this case, the log-linear oracle can be solved in closed form. In particular, we apply RFW to the computation of the Kacher mean and Wasserstein barycenters.