Title: Anarchic Federated Learning. Abstract: Present-day federated learning (FL) systems deployed over edge networks consists of a large number of workers with high degrees of heterogeneity in data and/or computing capabilities, which call for flexible worker participation in terms of timing, effort, data heterogeneity, etc. To achieve these goals, in this work, we propose a new FL paradigm called ``Anarchic Federated Learning'' (AFL). In stark contrast to conventional FL models, each worker in AFL has complete freedom to choose i) when to participate in FL, and ii) the number of local steps to perform in each round based on its current situation (e.g., battery level, communication channels, privacy concerns). However, AFL also introduces significant challenges in algorithmic design because the server needs to handle the chaotic worker behaviors. Toward this end, we propose two Anarchic Federated Averaging (AFA) algorithms with two-sided learning rates for both cross-device and cross-silo settings, which are named AFA-CD and AFA-CS, respectively. Somewhat surprisingly, even with general worker information arrival processes, we show that both AFL algorithms achieve the same convergence rate order as the state-of-the-art algorithms for conventional FL. Moreover, they retain the highly desirable {\em linear speedup effect} in the new AFL paradigm. We validate the proposed algorithms with extensive experiments on real-world datasets.