We introduce Episodic Memory QA, the task of answering personal user questions grounded on memory graph (MG), where episodic memories and related entity nodes are connected via relational edges. We create a new benchmark dataset first by generating synthetic memory graphs with simulated attributes, and by composing 100K QA pairs for the generated MG with bootstrapped scripts. To address the unique challenges for the proposed task, we propose Memory Graph Networks (MGN), a novel extension of memory networks to enable dynamic expansion of memory slots through graph traversals, thus able to answer queries in which contexts from multiple linked episodes and external knowledge are required. We then propose the Episodic Memory QA Net with multiple module networks to effectively handle various question types. Empirical results show improvement over the QA baselines in top-k answer prediction accuracy in the proposed task. The proposed model also generates a graph walk path and attention vectors for each predicted answer, providing a natural way to explain its QA reasoning.