In this paper, the first of two, we examine the problem of modal control of linear, time-invariant, multivariable descriptor systems. In particular, for systems satisfying a certain inequality we will give necessary and sufficient conditions for the existence of an output feedback matrix (possibly complex) which exactly assigns an arbitrary set of r (the number of independent semi-state variables of the system) distinct complex numbers to a generalized eigenspectrum of the closed loop system while ensuring that the resulting system has a unique smooth solution for each admissible initial condition. Furthermore, it is shown that if the set of proposed poles contains only real elements, the assignment can be carried out using real feedback. Next, we present conditions that are necessary and sufficient for pole assignability by real feedback of a seif-conjugate set of complex numbers provided that the system satisfies a further inequality.