This dissertation studies systems of "competing" discrete random walks as discrete and continuous time processes. A system is thought of as containing n imaginary particles performing random walks on lines parallel to the x-axis in Cartesian space. The particles act completely independently of each other and have, in general, different starting coordinates. In the discrete time situation, the motion of the n particles is governed by n independent streams of Bernoulli trials with success probabilities p₁, p₂,…, and p_n respectively. A success for any particle at a trial causes that particle to move one unit toward the origin, and a failure causes it to take a "zero-step" (i.e. remain stationary). A probabilistic description is first given of the positions of the particles at arbitrary points in time, and this is extended to provide time dependent and independent probabilities of which particle is the winner, that is to say, of which particle first reaches the origin. In this case "draws" are possible and the relevant probabilities are derived. The results are expressed, in particular, in terms of Generalized Hypergeometric Functions. In addition, formulae are given for the duration of what may now be regarded as a race with winning post at the origin. In the continuous time situation, the motion of the n particles is governed by n independent Poisson streams, in general, having different parameters. A treatment similar to that for the discrete time situation is given with the exception of draw probabilities which in this case are not possible. Approximations are obtained in many cases. Apart from their practical utility, these give insight into the operation of the systems in that they reveal how changes in one or more of the parameters may affect the win and draw probabilities and also the duration of the race. A chapter is devoted to practical applications. Here it is shown how the theory of random walks racing toward the origin can be utilized as a basic framework for explaining the operation of, and answering pertinent questions concerning several apparently diverse situations. Examples are Lanchester Combat theory, inventory control, reliability and queueing theory. / Ph. D.