  We prove explicit, i.e. non-asymptotic, error bounds for Markov chain Monte
Carlo methods. The problem is to compute the expectation of a function f with
respect to a measure {\pi}. Different convergence properties of Markov chains
imply different error bounds. For uniformly ergodic and reversible Markov
chains we prove a lower and an upper error bound with respect to the L2 -norm
of f . If there exists an L2 -spectral gap, which is a weaker convergence
property than uniform ergodicity, then we show an upper error bound with
respect to the Lp -norm of f for p > 2. Usually a burn-in period is an
efficient way to tune the algorithm. We provide and justify a recipe how to
choose the burn-in period. The error bounds are applied to the problem of the
integration with respect to a possibly unnormalized density. More precise, we
consider the integration with respect to log-concave densities and the
integration over convex bodies. By the use of the Metropolis algorithm based on
a ball walk and the hit-and-run algorithm it is shown that both problems are
polynomial tractable.
