Large language models (LLMs) distinguish themselves from previous technologies by functioning as collaborative ``thought partners,'' capable of engaging more fluidly in natural language on a range of tasks. As LLMs increasingly influence consequential decisions across diverse domains from healthcare to personal advice, the risk of overreliance -- relying on LLMs beyond their capabilities -- grows. This paper argues that measuring and mitigating overreliance must become central to LLM research and deployment. First, we consolidate risks from overreliance at both the individual and societal levels, including high-stakes errors, governance challenges, and cognitive deskilling. Then, we explore LLM characteristics, system design features, and user cognitive biases that together raise serious and unique concerns about overreliance on LLMs in practice. We also examine historical approaches for measuring overreliance, identifying three important gaps and proposing three promising directions to improve measurement. Finally, we propose mitigation strategies that can be pursued to ensure LLMs augment rather than undermine human capabilities.