T1 mapping is becoming a staple magnetic resonance imaging method for diagnosing myocardial diseases such as ischemic cardiomyopathy, hypertrophic cardiomyopathy, myocarditis, and more. Clinically, most T1 mapping sequences acquire a single slice at a single cardiac phase across a 10 to 15-heartbeat breath-hold, with one to three slices acquired in total. This leaves opportunities for improving patient comfort and information density by acquiring data across multiple cardiac phases in free-running acquisitions and across multiple respiratory phases in free-breathing acquisitions. Scanning in the presence of cardiac and respiratory motion requires more complex motion characterization and compensation. Most clinical mapping sequences use 2D single-slice acquisitions; however newer techniques allow for motion-compensated reconstructions in three dimensions and beyond. To further address confounding factors and improve measurement accuracy, T1 maps can be acquired jointly with other quantitative parameters such as T2, T2∗, fat fraction, and more. These multiparametric acquisitions allow for constrained reconstruction approaches that isolate contributions to T1 from other motion and relaxation mechanisms. In this review, we examine the state of the literature in motion-corrected and motion-resolved T1 mapping, with potential future directions for further technical development and clinical translation.