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Stereotypes are very harmful social constructs shaping human perception and behavior. Recent work shows that large language models (LLMs) may inherit and amplify such social harms. However, most existing research often focuses on stereotypical biases and overlooks stereotypes and the rich social-psychological literature on them, resulting in resource wastage and slowed progress in stereotype research. We argue that meaningful progress in mitigating stereotypes in LLMs requires tighter integration between social psychology and computational research. To address this gap, we review core social-psychological theories and frameworks and analyze their computational operationalization, highlighting substantial open opportunities. We also analyze computational progress across media narratives, body imaging, and multilingual, multicultural, and multimodal contexts, identifying key gaps and limitations in each domain. We also present a unified analysis of challenges in stereotype research. We further discuss implications for responsible AI, highlighting stereotypes as a root source of downstream harms, and briefly examine the limitations of current mitigation approaches along with potential improvements via explainability and interpretability. We frame stereotypes in AI as socio-technical phenomena and urge further research in responsible AI, informed by the perspectives and future directions presented in this paper.