In this thesis we have studied several aspects of the theory of Koopman and Perron-Frobenius operators associated with both discrete-time and continuous-time dynamical systems. The Perron-Frobenius operator describes the evolution of probability density governed by the underlying dynamical system, whereas the Koopman operator describes the evolution of observables, functions of the state space. Those operators are linear for any dynamical system, which allows us to describe nonlinear dynamical systems using a linear, yet infinite dimensional operator. The Perron-Frobenius operator and the Koopman operator have nice operator-theoretic properties: they are both partially isometric on their respective natural domains. Additionally, the Koopman operator associated with a transformation \(\Phi\) is the dual of the Perron-Frobenius operator associated with \(\Phi\), from which some spectral properties of the operators can be derived. For continuous-time dynamical systems we have the corresponding families, more specifically semigroups, of Koopman and Perron-Frobenius operators. If the dynamical system can be modeled by ordinary differential equations, the action of each family to certain classes of functions can be modeled using a single operator - the semigroup generator. The most popular method for approximating the Perron-Frobenius operator is Ulam’s method. The idea behind Ulam’s method is to split the state space into a finite number of disjoint sets (boxes) and model the system as a finite state Markov chain. Stationary distributions of that Markov chain approximate stationary densities of the Perron-Frobenius operator. The most widely used method for approximating the Koopman operator is Extended Dynamic Mode Decomposition (EDMD), which, given a dictionary of observables, approximates the projection of the Koopman operator onto the subspace spanned by those observables. A similar method, known as generator EDMD (gEDMD), can be employed to approximate the generator of the Koopman family. We have tested the efficiency of these methods on several examples of discrete-time and continuous-time dynamical systems.