This paper surveys foundation models for AI-enabled biological design, focusing on recent developments in applying large-scale, self-supervised models to tasks such as protein engineering, small molecule design, and genomic sequence design. Though this domain is evolving rapidly, this survey presents and discusses a taxonomy of current models and methods. The focus is on challenges and solutions in adapting these models for biological applications, including biological sequence modeling architectures, controllability in generation, and multi-modal integration. The survey concludes with a discussion of open problems and future directions, offering concrete next-steps to improve the quality of biological sequence generation.