A generalization of the description of a quantum system in the time interval between two measurements is presented. A new concept of a generalized state is introduced. Generalized states yield a complete description of a quantum system when information about the system is available both from the past and from the future. The formalism of generalized states provides a natural language for describing many peculiar situations. In particular, situations in which one can ascertain the result of a measurement of any one of several non-commuting variables are analysed. 'Weak' measurements on quantum systems described by generalized states are discussed. The relation between 'weak' and 'strong' measurements is investigated.