Attempts at discourse processing of spontaneously spoken dialogue face several difficulties: multiple hypotheses that result from the parsers attempts to make sense of the output from the speech recognizer, ambiguity that results from segmentation of multi-sentence utterances, and cumulative error - errors in the discourse context which cause further errors when subsequent sentences are processed. In this paper we will describe our robust parsers, our procedures for segmenting long utterances, and two approaches to discourse processing that attempt to deal with ambiguity and cumulative error.