Restricted normal modal logics are here defined by imposing conditions on the modal axioms and rules of normal modal systems. The conditions are defined in terms of a depth (associated with the modal connective) and a complexity function. It is proven that the logics obtained are characterized by a subtle adaptation of the possible worlds semantics in which levels are associated with the worlds. Restricted normal modal logics constitute a general framework allowing the definition of a huge variety of modal systems, which can have different applications. For instance, they are useful to define epistemic logics where the logical omniscience problem is partially controlled.