Artificial intelligence and neuromorphic computing driven by neural networks has enabled many applications. Software implementations of neural networks on electronic platforms are limited in speed and energy efficiency. Neuromorphic photonics aims to build processors in which optical hardware mimic neural networks in the brain. These processors promise orders of magnitude improvements in both speed and energy efficiency over purely digital electronic approaches. However, integrated optical neural networks are much smaller (hundreds of neurons) than electronic implementations (tens of millions of neurons). This raises a question: what are the applications where sub-nanosecond latencies and energy efficiency trump the sheer size of processor? We provide an overview of neuromorphic photonic systems and their real-world applications to machine learning and neuromorphic computing.