Random differential equations arise to model smooth random phenomena. The error term, instead of being introduced by means of a white noise, arises from imposing randomness to the input coefficients and initial/boundary conditions, with any distribution. We will establish theorems on existence and uniqueness of solution in the $L^p$ setting. We will focus on the first finite-dimensional distributions of the solution stochastic process, with two techniques: the Random Variable Transformation method and Karhunen-Loève expansions. When the probability density function of the response process cannot be computed, it is important to determine the expectation and variance at each time instant. We will give a summary on the main aspects of gPC expansions. The theory introduced in this thesis has permitted writing the following two articles: 10.1016/j.physa.2018.08.024 and 10.22436/jnsa.011.09.06 (article DOIs).