We propose the first general and practical framework to design certifiable algorithms for robust geometric perception in the presence of a large amount of outliers. We investigate the use of a truncated least squares (TLS) cost function, which is known to be robust to outliers, but leads to hard, nonconvex, and nonsmooth optimization problems. Our first contribution is to show that –for a broad class of geometric perception problems– TLS estimation can be reformulated as an optimization over the ring of polynomials and Lasserre’s hierarchy of convex moment relaxations is empirically tight at the minimum relaxation order (i.e., certifiably obtains the global minimum of the nonconvex TLS problem). Our second contribution is to exploit the structural sparsity of the objective and constraint polynomials and leverage basis reduction to significantly reduce the size of the semidefinite program (SDP) resulting from the moment relaxation, without compromising its tightness. Our third contribution is to develop scalable dual optimality certifiers from the lens of sums-of-squares (SOS) relaxation, that can compute the suboptimality gap and possibly certify global optimality of any candidate solution (e.g., returned by fast heuristics such as RANSAC or graduated non-convexity). Our dual certifiers leverage Douglas-Rachford Splitting to solve a convex feasibility SDP. Numerical experiments across different perception problems, including single rotation averaging, shape alignment, 3D point cloud and mesh registration, and high-integrity satellite pose estimation, demonstrate the tightness of our relaxations, the correctness of the certification, and the scalability of the proposed dual certifiers to large problems, beyond the reach of current SDP solvers.