This monograph is devoted to the study of some properties of the dynamics of stochastically excited differential equations of the form $$ \frac{\roman{d}}{\roman{d} t}X_j(t) = f_j(t,\bold{X}(t))+\sum_{\ell=1}^m g_{j\ell}(t,\bold{X}(t))\xi_\ell(t), \quad j=1,\ldots,n, $$ where $\bold{X}(t)=(X_1(t),\ldots,X_n(t))$ denotes the state variable and $\xi_1(t),\ldots,\xi_m(t)$ represent the system excitations, with at least one of these being a stochastic process. Some attention is given to the special cases of mechanical systems and Hamiltonian systems. \par Chapters 2 through 4 provide an overview of background material. More specifically, Chapter 2 contains a quick introduction to probability theory, Chapter 3 surveys key concepts and generalities related with stochastic processes and stochastic calculus, and Chapter 4 is focused on the relevant case of Markov processes. \par Chapter 5 is concerned with the analysis of linear systems subject solely to additive stochastic excitations. The authors provide some exact representations for the responses of these systems under such excitations, with special attention being given to the cases of stationary excitations and Gaussian excitations. Chapter 6 is devoted to the study of stochastically excited nonlinear dynamical systems and their stationary probability distributions. The authors discuss different procedures that may be used to obtain exact representations for such stationary distributions, namely, the methods of stationary potential, detailed balance and generalized stationary potential. The specific case of stochastically excited and dissipated Hamiltonian systems is discussed with further detail. Chapter 7 contains a discussion of approximation procedures that may be employed whenever the methods described in the previous chapter cannot be applied. Chapter 8 introduces concepts such as stochastic stability and stochastic bifurcations, and Chapter 9 is devoted to the analysis of the reliability of a stochastically excited system associated with the first-passage failure. \par All the topics included in the monograph are presented with a view towards applications, making it appealing to users without a strong background in theoretical mathematics. Thus, its main audience doesn't exactly seem to be the mathematical community, but instead researchers and graduate students from closely related scientific and technological subjects who want to explore the use of stochastic dynamics in their fields.