Recent developments in generative artificial intelligence (AI) rely on machine learning techniques such as deep learning and generative modeling to achieve state-of-the-art performance across wide-ranging domains. These methods' surprising performance is due in part to their ability to learn implicit "representations" of complex, multi-modal data. Unfortunately, deep neural networks are notoriously black boxes that obscure these representations, making them difficult to interpret or analyze. To resolve these difficulties, one approach is to build new interpretable neural network models from the ground up. This is the goal of the emerging field of causal representation learning (CRL) that uses causality as a vector for building flexible, interpretable, and transferable generative AI. CRL can be seen as a synthesis of three intrinsically statistical ideas: (i) latent variable models such as factor analysis; (ii) causal graphical models with latent variables; and (iii) nonparametric statistics and deep learning. This paper introduces CRL from a statistical perspective, focusing on connections to classical models as well as statistical and causal identifiability results. We also highlights key application areas, implementation strategies, and open statistical questions.