For linear autonomous multivariable control systems the conditions are studied under which the system is always--that is, for every control matrix of appropriate rank--completely controllable, which means that for every initial state one can construct a control which steers the system to any given final state in finite time. The resultant necessary and sufficient condition says that the number of independent controls either must equal the system's dimension or must be smaller by one. In the latter case no eigenvalue of the system matrix may be real. These results are valid for continuous systems as well as for sampled-data systems and are derived by means of well-known criteria for complete controllability and some theorems of matrix theory. Analogous results which are valid for the question of complete observability are formulated too. As a special case, previous results of Hsin Chu are covered.