Iwasawa theory is concerned with the behavior of arithmetic and algebro-geometric objects, such as ideal class groups and Mordell–Weil groups, in $\textbf{Z}_p$-extensions of number fields. The qualitative nature of the Iwasawa theory is dramatically different depending on whether the relevant Iwasawa modules are torsion or of positive rank, and the techniques needed to address these two disparate cases are accordingly different. While this principle was first perceived and studied over the cyclotomic $\textbf{Z}_p$-extension of a number field for supersingular elliptic curves, a similar one occurs for ordinary elliptic curves over the anticyclotomic $\textbf{Z}_p$-extension of an imaginary quadratic field. This talk will elaborate on this phenomenon and discuss some recent developments in the anticyclotomic setting.