The anticyclotomic main conjectures for elliptic curves

Let $E/\mathbf{Q}$ be an elliptic curve of conductor N and let f be the cuspidal eigenform on $\Gamma_0(N)$ associated to E by the modularity theorem. Denote by $K_\infty$ the anticyclotomic p-extension of an imaginary quadratic field K, where p is a prime number unramified in K. Under appropriate arithmetic assumptions we prove the main conjectures of Iwasawa theory for E over $K_\infty$. Our results cover both the cases where p is good ordinary and supersingular for E, and both the definite and indefinite settings. This leaves out a single case, which we term exceptional, for which we establish one of the two expected divisibilities.