We characterize a class of Markovian dynamics using the concept of divisible dynamical map.Moreover we provide a family of criteria which can distinguish Markovian and non-Markovian dynamics.These "Markovianity criteria" are based on a simple observation that Markovian dynamics implies monotonic behavior of several well known quantities like distinguishability of states,fidelity,relative entropy and genuine entanglement measures.Finally,we compare the notion of Markovianity based on divisibility with that based on distinguishability of quantum states.The talk is illustrated by simple examples of qubit dynamics.It is shown how the symmetry properties of the corresponding time-dependent generator influences the dynamics of the quantum system.