<p>The rapidly increasing demands on computational throughput, bandwidth and memory capacity fuelled by breakthroughs in machine learning pose substantial challenges for conventional electronic computing platforms. Historically, advancing compute performance relied on miniaturization to increase the transistor count on a given chip area and, more recently, on the development of parallel and multicore architectures. Computing platforms that process data using multiple, orthogonal dimensions can achieve exponential scaling on trajectories much steeper than what is possible with conventional strategies. One promising analog platform is photonics, which makes use of the physics of light, such as sensitivity to material properties and ability to encode information across multiple degrees of freedom. With recent breakthroughs in integrated photonic hardware and control, large-scale photonic systems have become a practical and timely solution for data-intensive, real-time computational tasks. Here, we explain developments in the realization of multidimensional computing platforms based on photonic systems. Moving to such architectures holds promise for low-latency, high-bandwidth information processing at reduced energy consumption.</p>