An axiomatic theory of hierarchical systems is presented. This theory is intended to apply to general multilevel systems and to provide a framework for mathematical modeling of such systems. As an example, several possible implications for Miller's (1978) living systems theory are mentioned. The mathematical development of the general theory is continued by imposing a restriction of vertical linearity. Several theorems relating to issues of control are proved for vertically linear systems. A theorem showing the feasibility of environmental interaction via cyclic time-lagged correlations between system variables is proved.