Deep artificial neural networks apply principles of the brain's information processing that led to breakthroughs in machine learning spanning many problem domains. Neuromorphic computing aims to take this a step further to chips more directly inspired by the form and function of biological neural circuits, so they can process new knowledge, adapt, behave, and learn in real time at low power levels. Despite several decades of research, until recently, very few published results have shown that today's neuromorphic chips can demonstrate quantitative computational value. This is now changing with the advent of Intel's Loihi, a neuromorphic research processor designed to support a broad range of spiking neural networks with sufficient scale, performance, and features to deliver competitive results compared to state-of-the-art contemporary computing architectures. This survey reviews results that are obtained to date with Loihi across the major algorithmic domains under study, including deep learning approaches and novel approaches that aim to more directly harness the key features of spike-based neuromorphic hardware. While conventional feedforward deep neural networks show modest if any benefit on Loihi, more brain-inspired networks using recurrence, precise spike-timing relationships, synaptic plasticity, stochasticity, and sparsity perform certain computation with orders of magnitude lower latency and energy compared to state-of-the-art conventional approaches. These compelling neuromorphic networks solve a diverse range of problems representative of brain-like computation, such as event-based data processing, adaptive control, constrained optimization, sparse feature regression, and graph search.