Simplicial complexes are generalized network structures able to encode interactions occurring between more than two nodes. Simplicial complexes describe a large variety of complex interacting systems ranging from brain networks to social and collaboration networks. Here we characterize the structure of simplicial complexes using their generalized degrees that capture fundamental properties of one, two, three, or more linked nodes. Moreover, we introduce the configuration model and the canonical ensemble of simplicial complexes, enforcing, respectively, the sequence of generalized degrees of the nodes and the sequence of the expected generalized degrees of the nodes. We evaluate the entropy of these ensembles, finding the asymptotic expression for the number of simplicial complexes in the configuration model. We provide the algorithms for the construction of simplicial complexes belonging to the configuration model and the canonical ensemble of simplicial complexes. We give an expression for the structural cutoff of simplicial complexes that for simplicial complexes of dimension d=1 reduces to the structural cutoff of simple networks. Finally, we provide a numerical analysis of the natural correlations emerging in the configuration model of simplicial complexes without structural cutoff.