Unsupervised domain adaptation (UDA) approaches focus on adapting models trained on a labeled source domain to an unlabeled target domain. In contrast to UDA, source-free domain adaptation (SFDA) is a more practical setup as access to source data is no longer required during adaptation. Recent state-of-the-art (SOTA) methods on SFDA mostly focus on pseudo-label refinement based self-training which generally suffers from two issues: i) inevitable occurrence of noisy pseudo-labels that could lead to early training time memorization, ii) refinement process requires maintaining a memory bank which creates a significant burden in resource constraint scenarios. To address these concerns, we propose C-SFDA, a curriculum learning aided self-training framework for SFDA that adapts efficiently and reliably to changes across domains based on selective pseudo-labeling. Specifically, we employ a curriculum learning scheme to promote learning from a restricted amount of pseudo labels selected based on their reliabilities. This simple yet effective step successfully prevents label noise propagation during different stages of adaptation and eliminates the need for costly memory-bank based label refinement. Our extensive experimental evaluations on both image recognition and semantic segmentation tasks confirm the effectiveness of our method. C-SFDA is also applicable to online test-time domain adaptation and outperforms previous SOTA methods in this task.