We present our recent work (ICS 2011) on dynamic environments in which computational nodes, or decision makers, follow simple and unsophisticated rules of behavior (e.g., repeatedly "best replying" to others' actions, and minimizing "regret") that have been extensively studied in game theory and economics. We aim to understand when convergence of the resulting dynamics to an equilibrium point is guaranteed if nodes' interaction is not synchronized (e.g., as in Internet protocols and large-scale markets). We take the first steps of this research agenda. We exhibit a general non-convergence result and consider its implications across a wide variety of interesting and timely applications: routing, congestion control, game theory, social networks and circuit design. We also consider the relationship between classical nontermination results in distributed computing theory and our result, explore the impact of scheduling on convergence, study the computational and communication complexity of asynchronous dynamics and present some basic observations regarding the effects of asynchrony on no-regret dynamics.