Quantum quench is a way of creating nonthermal excitations of various quantum systems. It consists in a sudden change of control parameters between selected initial and final points in the parameter space for the system prepared in the ground state at the initial point. The quench-induced dynamics makes it possible to investigate general features of the equilibration processes in the quantum domain. The evolution of the survival probability of the initial state after the quench manifests surprisingly rich structures and reveals several nontrivial dynamical stages. We will show that quantum quench protocols can be tailored to probe dynamical response of the system in selected excitation regions, disclosing distinct quantum phases in the spectrum of stationary states and detecting transitions between these phases. We will also discuss a novel type of criticality connected with the time variable. Showing up as nonanalyticities in the time dependence of various observables in the infinite-size limit, these so-called dynamical quantum phase transitions represent an interesting analogy of thermal phase transitions and can be studied by similar theoretical techniques.