We present two logics of collective belief with a semantics exploiting the notion of belief base. The semantics distinguishes explicit from implicit belief: an agent's belief of explicit type is a piece of information contained in the agent's belief base, while a belief of implicit type corresponds to a piece of information that is derivable from the agent's belief base. The first part of the paper is devoted to the logic of implicit common belief, while the second presents the logic of explicit common belief. Implicit common belief is defined as a mutual belief of any order. This leads to the usual fixpoint construction of common belief. Explicit common belief is the collective counterpart of explicit individual belief and has a public nature. It moreover implies implicit common belief. We study axiomatic aspects of our logics as well as complexity of satisfiability checking. We show that, while the satisfiability checking problem is EXPTIME-hard for the logic of implicit common belief, it is in PSPACE for the logic of explicit common belief. This makes the latter logic a natural candidate for reasoning about collective attitudes in multi-agent scenarios and applications. We also study a dynamic extension of the logic of explicit common belief in which private and public forms of information dynamics can be modeled.