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In order to meet the requirements for performance, safety, and latency in many IoT applications, intelligent decisions must be made right here right now at the network edge. However, the constrained resources and limited local data amount pose significant challenges to the development of edge AI. To overcome these challenges, we explore continual edge learning capable of leveraging the knowledge transfer from previous tasks. Aiming to achieve fast and continual edge learning, we propose a platform-aided federated meta-learning architecture where edge nodes collaboratively learn a meta-model, aided by the knowledge transfer from prior tasks. The edge learning problem is cast as a regularized optimization problem, where the valuable knowledge learned from previous tasks is extracted as regularization. Then, we devise an ADMM based federated meta-learning algorithm, namely ADMM-FedMeta, where ADMM offers a natural mechanism to decompose the original problem into many subproblems which can be solved in parallel across edge nodes and the platform. Further, a variant of inexact-ADMM method is employed where the subproblems are `solved' via linear approximation as well as Hessian estimation to reduce the computational cost per round to $\mathcal{O}(n)$. We provide a comprehensive analysis of ADMM-FedMeta, in terms of the convergence properties, the rapid adaptation performance, and the forgetting effect of prior knowledge transfer, for the general non-convex case. Extensive experimental studies demonstrate the effectiveness and efficiency of ADMM-FedMeta, and showcase that it substantially outperforms the existing baselines.