Artificial intelligence (AI) technologies have dramatically advanced in recent years, resulting in revolutionary changes in people's lives. Empowered by edge computing, AI workloads are migrating from centralized cloud architectures to distributed edge systems, introducing a new paradigm called edge AI. While edge AI has the promise of bringing significant increases in autonomy and intelligence into everyday lives through common edge devices, it also raises new challenges, especially for the development of its algorithms and the deployment of its services, which call for novel design methodologies catered to these unique challenges. In this paper, we provide a comprehensive survey of the latest enabling design methodologies that span the entire edge AI development stack. We suggest that the key methodologies for effective edge AI development are single-layer specialization and cross-layer co-design. We discuss representative methodologies in each category in detail, including on-device training methods, specialized software design, dedicated hardware design, benchmarking and design automation, software/hardware co-design, software/compiler co-design, and compiler/hardware co-design. Moreover, we attempt to reveal hidden cross-layer design opportunities that can further boost the solution quality of future edge AI and provide insights into future directions and emerging areas that require increased research focus.