The hallmark of effective language use lies in consistency: expressing similar meanings in similar contexts and avoiding contradictions. While human communication naturally demonstrates this principle, state-of-the-art language models (LMs) struggle to maintain reliable consistency across task- and domain-specific applications. Here we examine the landscape of consistency research in LMs, analyze current approaches to measure aspects of consistency, and identify critical research gaps. Our findings point to an urgent need for quality benchmarks to measure and interdisciplinary approaches to ensure consistency while preserving utility.