This chapter first reviews the basic architectural concepts that underlie the formal theory of Lexical-Functional Grammar. The LFG formalism provides a simple set of devices for describing the common properties of all human languages and the particular properties of individual languages. It postulates two levels of syntactic representation for a sentence, a constituent structure and a functional structure. These are related by a piecewise correspondence that permits the abstract functional structure to be described in terms of configurations of constituent structure phrases. We then survey the mathematical and computational properties of this simple framework. We demonstrate that the recognition/parsing, realization/generation, emptiness, and other more specific decision problems are unsolvable for grammars in the unrestricted LFG formalism. A first set of restrictions guarantees decidability of recognition, realization, and other problems for grammars that are still suitable for linguistic description, but the solutions to these problems in the worst case are computationally impractical. The class of LFG grammars that meet an additional set of restrictions is equivalent to the class of mildly context-sensitive grammars, and the recognition and realization problems for grammars in this class are thus not only decidable but tractable as well.