This paper is primarily a survey of theory developed in the early and mid 1970's by H. O. Kreiss and others. The theory discussed deals with boundary-initial value problems for linear first order hyperbolic systems. A brief discussion is given in one space dimension. Here it is shown that boundary conditions may not be chosen arbitrarily since portions of the boundary may represent travelling waves. After this, multi-dimensional problems are considered, where it is more difficult to determine whether a given boundary condition is suitable for a given equation. An account of the ''energy method'' is included, which gives criteria sufficient for a boundary condition to yield a well-posed problem. Summarizing work by Kreiss, Sakamoto, Rauch, Ralston, Majda, Osher and Michelson, the ideas of ''normal model analysis'' are then reviewed, this theory giving criteria that are essentially necessary and sufficient for a boundary condition to yield a well-posed problem. Since the theory is complicated, the clarification and concise presentation of the theory is one of the main purposes of the paper. In particular the paper interprets clearly criteria that determine whether a boundary condition is suitable. A main theme is to concentrate on the idea of a ''characteristic variety'' of the system.