In this paper, a random clique network model to mimic the large clustering coefficient and the modular structure that exist in many real complex networks, such as social networks, artificial networks, and protein interaction networks, is introduced by combining the random selection rule of the Erdös and Rényi (ER) model and the concept of cliques. We find that random clique networks having a small average degree differ from the ER network in that they have a large clustering coefficient and a power law clustering spectrum, while networks having a high average degree have similar properties as the ER model. In addition, we find that the relation between the clustering coefficient and the average degree shows a non-monotonic behavior and that the degree distributions can be fit by multiple Poisson curves; we explain the origin of such novel behaviors and degree distributions.