I will discuss several probabilistic models producing simplicial complexes, manifolds and discrete groups. Random simplicial complexes are high dimensional analogues of random graphs and can be used for studying the behaviour of large systems or networks depending on many random parameters. We are interested in properties of random spaces which are satisfies with probability tending to one. Using probabilistic models one may also test probabilistically the validity of open topological questions such as the Whitehead and the Eilenberg – Ganea conjectures.