Simplicial complexes are generalized network structures able to encode interactions occurring between more than two nodes. Simplicial complex describe a large variety of complex interacting systems ranging from brain networks, to social and collaboration networks. Additionally simplicial complexes have a geometrical interpretation and for this reasons they have been widely used in quantum gravity. Simplicial complexes are the ideal structures to characterize emergent network geometry in which geometrical properties of the networks emerge spontaneously from their dynamics. Here we propose a general model for growing simplicial complexes called network geometry with flavor (NGF) and we characterize the configuration model of simplicial complexes. These models deepens our understanding of complex networks and reveals the important effect that the dimensionality of growing simplicial complexes have on their structure. The NGF can generate discrete geometries of different nature, ranging from chains and higher dimensional manifolds to scale-free networks with small-world properties, scale-free degree distribution and non-trivial community structure. We find that, for NGF with dimension greater than one, scale-free topologies emerge also without including an explicit preferential attachment because and efficient preferential attachment mechanism naturally emerges from the dynamical rules. Interestingly the NGF with fitness of the nodes reveals relevant relations with quantum statistics. The configuration model of simplicial complexes characterizes instead static simplicial complexes. Here the ensemble will be discussed highlighting the differences with the configuration model of networks, showing that the effect of the increased dimesionality of simplicial complexes is reflected in the different structural cutoff and in the specific nature of the degree correlations.