We discuss the behaviour of dynamical systems perturbed by small Levy noise with heavy tails. In particular we determine the asymptotics of first exit times for systems with a) additive noise with power tails, b) additive noise with sub- and super-exponential tails; c) multiplicative noises of Ito, Stratonovich and Marcus type. As examples we consider $\alpha$-stable Levy ratchet and a motion of a charged Levy particle in an external magnetic field.