  We present the first accelerated randomized algorithm for solving linear
systems in Euclidean spaces. One essential problem of this type is the matrix
inversion problem. In particular, our algorithm can be specialized to invert
positive definite matrices in such a way that all iterates (approximate
solutions) generated by the algorithm are positive definite matrices
themselves. This opens the way for many applications in the field of
optimization and machine learning. As an application of our general theory, we
develop the {\em first accelerated (deterministic and stochastic) quasi-Newton
updates}. Our updates lead to provably more aggressive approximations of the
inverse Hessian, and lead to speed-ups over classical non-accelerated rules in
numerical experiments. Experiments with empirical risk minimization show that
our rules can accelerate training of machine learning models.
