This thesis derives the theory of distributions, starting with test functions as a basis. Distributions and their derivatives will be analysed and exemplified. Schwartz functions are introduced, and the Fourier transform of Schwartz functions is analysed, creating the basis for Tempered distributions on which we also analyse the Fourier transform. Weak derivatives and Sobolev spaces are defined, and from the Fourier transform we define Sobolev spaces of non-integer order. The theory presented is applied to an initial value problem with a derivative of order one in time and an arbitrary differentiation operator in space, and we take a look at conditions for well-posedness under different differnetiation operators and present some minor results. The Riesz representation theorem and the Lax--Milgram theorem are presented in order to offer a different perspective on the results from the initial value problem.