This article establishes several equiconsistency results for the three set theories \roster \item"(A)" $\ssf{ZF}+\ssf{DC}+\ssf{AD}$, \item"(B)" $\ssf{ZF}+\ssf{AD}_\Bbb{R}$, and \item"(C)" $\ssf{ZF}+\ssf{DC}+\ssf{AD}_\Bbb{R}$ \endroster with theories involving notions of strong compactness. Here, $\ssf{AD}$ and $\ssf{AD}_\Bbb{R}$ are the axioms of determinacy for natural numbers and reals and $\ssf{DC}$ is the axiom of dependent choice. \par For an uncountable set $X$, $\omega_1$ is $X$-{\it strongly compact} if and only if there is a measure (ultrafilter) $\mu$ on $\wp_{\omega_1}(X)$ that is countably complete and fine: $\mu$ contains the set $$ \{\sigma\in\wp_{\omega_1}(X):x\in\sigma\} $$ for all $x\in X$. \par The main results are as follows. \par (A) is equiconsistent with each of the following: \roster \item"(A$_1$)" $\ssf{ZF}+\ssf{DC}$ + $\omega_1$ is $\wp(\omega_1)$-strongly compact, \item"(A$_2$)" $\ssf{ZF}+\ssf{DC} + \omega_1$ is $\Bbb{R}$-strongly compact and $\omega_2$-strongly compact, and \item"(A$_3$)" $\ssf{ZF}+\ssf{DC} + \omega_1$ is $\Bbb{R}$-strongly compact and $\lnot\square_{\omega_1}$. \endroster \par (B) is equiconsistent with \roster \item"(B$_1$)" $\ssf{ZF}+\ssf{DC}_{\wp(\omega_1)}+ \omega_1$ is $\Bbb{R}$-strongly compact and $\Theta$ is singular. \endroster \par (C) is equiconsistent with each of the following: \roster \item"(C$_1$)" $\ssf{ZF}+\ssf{DC}$ + $\omega_1$ is $\wp(\Bbb{R})$-strongly compact, \item"(C$_2$)" $\ssf{ZF}+\ssf{DC}$ + $\omega_1$ is $\Bbb{R}$-strongly compact and $\Theta$-strongly compact, and \item"(C$_3$)" $\ssf{ZF}+\ssf{DC}$ + $\omega_1$ is $\Bbb{R}$-strongly compact and $\Theta$ is singular. \endroster \par Here, $\ssf{DC}_{\wp(\omega_1)}$ is the fragment of dependent choice that allows us to choose $\omega$-sequences from $\wp(\omega_1)$ and $\Theta$ is the smallest ordinal that is not a surjective image of $\Bbb{R}$. \par The proofs involve a good deal of fine structure and descriptive inner model theory and build heavily on many earlier results of the authors and others.