Deep learning has revolutionized many sectors of industry and daily life, but as application scale increases, performing training and inference with large models on massive datasets is increasingly unsustainable on existing hardware. Highly parallelized hardware like Graphics Processing Units (GPUs) are now widely used to improve speed over conventional Central Processing Units (CPUs). However, Complementary Metal-oxide Semiconductor (CMOS) devices suffer from fundamental limitations relying on metallic interconnects which impose inherent constraints on bandwidth, latency, and energy efficiency. Indeed, by 2026, the projected global electricity consumption of data centers fueled by CMOS chips is expected to increase by an amount equivalent to the annual usage of an additional European country. Silicon Photonics (SiPh) devices are emerging as a promising energy-efficient CMOS-compatible alternative to electronic deep learning accelerators, using light to compute as well as communicate. In this review, we examine the prospects of photonic computing as an emerging solution for acceleration in deep learning applications. We present an overview of the photonic computing landscape, then focus in detail on SiPh integrated circuit (PIC) accelerators designed for different neural network models and applications deep learning. We categorize different devices based on their use cases and operating principles to assess relative strengths, present open challenges, and identify new directions for further research.