Recent advances in bioengineering have enabled the creation of biological neural networks in vitro, raising the prospect of novel, unconventional platforms that can leverage genuine biological computation. The technology could help unlock computing paradigms that could be faster, more powerful, and more energy efficient than the silicon-based architectures that dominate today's computing landscape. However, engineering cell cultures for computing applications presents a radical departure from digital von Neumann architectures that computer scientists have grown accustomed to and will require a rethink of the entire stack. Here, we provide a brief overview of the key technologies, principles, and challenges of this emerging interdisciplinary field. We argue that seizing on its potential will require the development of new machine-learning approaches that can process the vast observable activity of neuronal cell cultures and learn to control and make sense of their neural code. Such an effort could provide a pathway for leveraging biological neural networks and contribute to our understanding of what makes biological learning in neurons so incredibly efficient, holding broader lessons for the development of next-generation AI systems.