This thesis studies the problem of estimating the largest possible dimension of a linear space of real matrices under the assumption that every non-zero matrix in the space has (the same) fixed rank. The complex version of this problem has been studied by R. Westwick and J. Sylvester. Sylvester introduced a technique based on the theory of Chern classes for estimating the dimension from above. The question of determining the largest dimension of a linear space of maximal-rank real n x n matrices (or, equivalently, of determining the largest number of nonsingular n x n matrices all of whose non-trivial linear combinations are non-singular) was solved by J.F. Adams, P. Lax and R. Phillips. Their proof uses Adams' solution of the vector fields on spheres problem to show that the linear spaces constructed by J. Radon and A. Hurwitz are of the largest possible dimension under this hypothesis. A number of general results on the dimensions of linear spaces of fixed-rank real matrices, as well as related questions concerning linear spaces whose non-zero matrices have rank bounded below, are due to E. Rees and K.Y. Lam. The method used to provide upper bounds for the dimension is analogous to the complex case; here Stiefel-Whitney classes and K-theory are used for the calculations. Clifford Algebras are then used to construct spaces and so provide lower bounds for the dimension. We show how calculations with Stiefel-Whitney classes together with information about the existence of certain bilinear maps enable us to determine the dimensions of spaces of real n x k matrices of fixed-rank k for all n and k with k ≤ 9. The case of fixed-rank symmetric matrices is also investigated. The main result here is that every space of real symmetric n x n matrices of fixed rank 2k + 1 must have dimension 1.