The random walk in continuous time has been considered by a number of authors for particular types of boundary conditions, e. g. the unrestricted random walk by Irwin (1937); the random walk with one absorbing or one reflecting barrier by Ledermann & Reuter (1954), and subsequently by Bailey (1954). The present paper gives a unified theory of these processes which includes previous work on the subject as special cases and readily yields the solution for any other consistent form of boundary conditions. Several of these are of interest in the theory of queues with exponential service and arrival times. The random walk with reflecting barriers at Oand N (§ 4) is the single server queue problem where the number of possible customers is limited to N (Morse, 1958). By letting N → ∞ we obtain the familiar single server queue problem solved previously in the papers by Ledermann & Reuter and by Bailey referred to above and also by Clarke (1956), Champernowne (1956) and Conolly (1958). The passage to a ‘diffusion’ process is also considered. The method given provides an alternative to the well-known method of images (Chandrasekhar, 1943; Bartlett, 1956). In § 5 we consider a more general process where the transition intensities λ and μ depend linearly on the particle‘s position in a bounded interval, and are constant outside this interval. This enables us to give the Laplace transform of the probability generating function for the N server queue, a problem recently considered by Karlin & McGregor (1958).