  A universal definition of non-Markovianity for open systems dynamics is
proposed. It is extended from the classical definition to the quantum realm by
showing that a `transition' from the Markov to the non-Markov regime occurs
when the correlations between the system and the environment, generated by
their joint evolution, can no longer be neglected. The suggested definition is
based on the comparison between measured correlation functions and those built
by assuming that the system is in a Markov regime thus giving a figure of merit
of the error coming from this assumption. It is shown that the knowledge of the
dynamical map and initial condition of the system is not enough to fully
characterise the non-Markovian dynamics of the reduced system. The example of
three exactly solvable models, i.e. decoherence and spontaneous emission of the
qubit in a bosonic bath and decoherence of the photon's polarization induced by
interaction with its spacial degrees of freedom, reveals that previously
proposed Markovianity criteria and measures which are based on dynamical map
analysis fail to recognise non-Markov behaviour.
