Exact and approximate solutions of the initial-boundary value problem for the Korteweg-de Vries equation on the semi-infinite line are found. These solutions are found for both constant and time-dependent boundary values. The form of the solution is found to depend markedly on the specific boundary and initial value. In particular, multiple solutions and nonsteady solutions are possible. The analytical solutions are compared with numerical solutions of the Korteweg-de Vries equation and are found to be in good agreement. © 1991 Oxford University Press.