The concepts of controllability and strong controllability and some other closely related concepts like reachability and irreducibility that appear in the theory of nonsquare implicit systems are considered. The main differences between these concepts in the cases of non-square and regularizable systems, are discussed and their new definitions are introduced. Then, using the feedback canonical form of the system, necessary and sufficient conditions of controllability and strong controllability are derived in terms of its feedback invariants, and geometric terms, respectively.