  A measure on a locally compact group is called spread out if one of its
convolution powers is not singular with respect to Haar measure. Using Markov
chain theory, we conduct a detailed analysis of random walks on homogeneous
spaces with spread out increment distribution. For finite volume spaces, we
arrive at a complete picture of the asymptotics of the $n$-step distributions:
They equidistribute towards Haar measure, often exponentially fast and locally
uniformly in the starting position. In addition, many classical limit theorems
are shown to hold. In the infinite volume case, we prove recurrence and a ratio
limit theorem for symmetric spread out random walks on homogeneous spaces of at
most quadratic growth. This settles one direction in a long-standing
conjecture.
