Superconformal field theories placed in nontrivial background fields for the metric and the continuous global symmetries exhibit a rich web of RG flows across dimensions. I will discuss several examples of such flows and emphasize some of their universal features. In addition, I will employ non-perturbative tools such as 't Hooft anomaly matching, a-, F-, and c-extremization, and holography to gain a quantitative understanding of some aspects of these theories. Finally, I will discuss the relevance of these results for a microscopic understanding of the entropy of supersymmetric black holes and strings in AdS.