Linear deterministic and stochastic control systems are considered. Different types of controllability are defined and discussed. Using methods taken from functional analysis, several conditions for approximate controllability, complete controllability and stochastic controllability are formulated and proved. The relations between different types of controllability are explained and discussed. Special attention is paid to the role of the operator Riccati equation in controllability investigations. A simple illustrative example is also presented. Moreover, several remarks and comments on controllability problems for infinite-dimensional control systems are given. Similar problems have been recently considered in the paper [A. E. Bashirov and K. R. Kerimov, SIAM J. Control Optimization 35, No. 2, 384-398 (1997; Zbl 0873.93076)].