We introduce a new semantics for a family of logics of explicit and implicit belief based on the concept of multi-agent belief base. Differently from standard semantics for epistemic logic in which the notions of possible world and doxastic/epistemic alternative are primitive, in our semantics they are non-primitive but are computed from the concept of belief base. We provide complete axiomatizations and prove decidability for our logics via finite model arguments. Furthermore, we provide polynomial embeddings of our logics into Fagin & Halpern's logic of general awareness and establish complexity results via the embeddings. We also present variants of the logics incorporating different forms of epistemic introspection for explicit and/or implicit belief and provide complexity results for some of these variants. Finally, we present a number of dynamic extensions of the static framework by informative actions of both public and private type, including public announcement, belief base expansion and forgetting. We illustrate the application potential of the logical framework with the aid of a concrete example taken from the domain of conversational agents.