Multilingual pre-trained language models are incredibly effective at Question Answering (QA), a core task in Natural Language Understanding, achieving high accuracies on several multilingual benchmarks. However, little is known about how well they are calibrated. In this paper, we study the calibration properties of several pre-trained multilingual large language models (LLMs) on a variety of question-answering tasks. We perform extensive experiments, spanning both extractive and generative QA model designs and diverse languages, spanning both high-resource and low-resource ones. We study different dimensions of calibration in in-distribution, out-of-distribution, and cross-lingual transfer settings, and investigate strategies to improve it, including post-hoc methods and regularized fine-tuning. We demonstrate automatically translated data augmentation as a highly effective technique to improve model calibration. We also conduct a number of ablation experiments to study the effect of model size on calibration and how multilingual models compare with their monolingual counterparts for diverse tasks and languages.