A successful negotiation demands a deep comprehension of the conversation context, Theory-of-Mind (ToM) skills to infer the partner's motives, as well as strategic reasoning and effective communication, making it challenging for automated systems. Given the remarkable performance of LLMs across a variety of NLP tasks, in this work, we aim to understand how LLMs can advance different aspects of negotiation research, ranging from designing dialogue systems to providing pedagogical feedback and scaling up data collection practices. To this end, we devise a methodology to analyze the multifaceted capabilities of LLMs across diverse dialogue scenarios covering all the time stages of a typical negotiation interaction. Our analysis adds to the increasing evidence for the superiority of GPT-4 across various tasks while also providing insights into specific tasks that remain difficult for LLMs. For instance, the models correlate poorly with human players when making subjective assessments about the negotiation dialogues and often struggle to generate responses that are contextually appropriate as well as strategically advantageous.