This thesis presents and tests an incremental theory of Lexical Functional Grammar (LFG) in an attempt to support researchers in formal grammar to engage with questions raised by experimental findings on language processing and artificial intelligence. Previous work on the incremental building of syntactic structure has concentrated on constituent structure, considering interactions between word class and phrase-structure rules. However, syntax in LFG is represented not only as constituent structure, but also as functional structure: a universal set of grammatical functions is included within the primitives of the theory. The incremental theory presented here explores the role of grammatical functions as well as category and phrase-structure rules, to build representations of c-structure and f-structure. Universal and language-specific well-formedness constraints interact with lexical content to shape the process of incremental structural growth. The theory is then used to derive predictions about the impact of context on processing decisions. The theory is tested computationally using a model built in the ACT-R computational cognitive architecture. The model's production set assesses the combinatorial constraints that apply to lexical input in the context of previously processed material and builds structure accordingly. The combinatorial constraints are contained in lexical specifications stored in declarative, rather than procedural memory. As a result of this the model can use a single production set to process lexical input in English and Korean. The outputs of simulations show that the model is capable of building monoclausal and complex sentences in the two language. The model also simulates context-dependent variations in the parsing of identical strings. A language-independent proposal of the effect of prosodic breaks is included in the model, and this is used to simulate prosodic facilitation of the processing of complex syntax.