Recently, model-agnostic meta-learning (MAML) has garnered tremendous attention. However, stochastic optimization of MAML is still immature. Existing algorithms for MAML are based on the ``episode" idea by sampling a number of tasks and a number of data points for each sampled task at each iteration for updating the meta-model. However, they either do not necessarily guarantee convergence with a constant mini-batch size or require processing a larger number of tasks at every iteration, which is not viable for continual learning or cross-device federated learning where only a small number of tasks are available per-iteration or per-round. This paper addresses these issues by (i) proposing efficient memory-based stochastic algorithms for MAML with a diminishing convergence error, which only requires sampling a constant number of tasks and a constant number of examples per-task per-iteration; (ii) proposing communication-efficient distributed memory-based MAML algorithms for personalized federated learning in both the cross-device (w/ client sampling) and the cross-silo (w/o client sampling) settings. The key novelty of the proposed algorithms is to maintain an individual personalized model (aka memory) for each task besides the meta-model and only update them for the sampled tasks by a momentum method that incorporates historical updates at each iteration. The theoretical results significantly improve the optimization theory for MAML and the empirical results also corroborate the theory.