Flows on normed spaces can be classified using flow equivalences --- maps on the space with the property that the structure of one flow is converted into the structure of another flow. Of particular interest are classifications that arise from flow equivalences that are either homeomorphisms or diffeomorphisms. It is possible to completely characterize such classifications based solely on a few simple properties of flows, at least in the case of linear flows on finite-dimensional normed spaces. Results concerning diffeomorphic classification are well known and can be found in many textbooks that discuss continuous dynamical systems. The situation is similar when it comes to homeomorphic classification of hyperbolic flows, but for arbitrary (possibly nonhyperbolic) flows results concerning homeomorphic classification are fairly obscure. This thesis aims to provide a complete discussion of the homeomorphic and diffeomorphic classification of linear flows on finite-dimensional normed spaces.