Hit-and-Run is known to be one of the best versions of Markov Chain Monte Carlo sampler. Nevertheless, in practice the number of iterations required to achieve uniformly distributed samples is rather high. We propose new random walk algorithm based on billiard trajectories and prove its asymptotic uniformity. Numerical experiments demonstrate much faster convergence to uniform distribution for Billiard Walk algorithm compared to Hit-and- Run. We discuss a class of global optimization problems that can be efficiently solved with Monte Carlo sampler.