While Moore's law has driven exponential computing power expectations, its nearing end calls for new avenues for improving the overall system performance. One of these avenues is the exploration of new alternative brain-inspired computing architectures that promise to achieve the flexibility and computational efficiency of biological neural processing systems. Within this context, neuromorphic intelligence represents a paradigm shift in computing based on the implementation of spiking neural network architectures tightly co-locating processing and memory. In this paper, we provide a comprehensive overview of the field, highlighting the different levels of granularity present in existing silicon implementations, comparing approaches that aim at replicating natural intelligence (bottom-up) versus those that aim at solving practical artificial intelligence applications (top-down), and assessing the benefits of the different circuit design styles used to achieve these goals. First, we present the analog, mixed-signal and digital circuit design styles, identifying the boundary between processing and memory through time multiplexing, in-memory computation and novel devices. Next, we highlight the key tradeoffs for each of the bottom-up and top-down approaches, survey their silicon implementations, and carry out detailed comparative analyses to extract design guidelines. Finally, we identify both necessary synergies and missing elements required to achieve a competitive advantage for neuromorphic edge computing over conventional machine-learning accelerators, and outline the key elements for a framework toward neuromorphic intelligence.