The large N limit of confined-phase 4D gauge theories is a free limit in terms of the physical degrees of freedom, the mesons and glueballs. But the resulting free theories have not been solved despite 40 years of attempts, essentially due to the difficulty of finding some principle that would usefully constrain the spectral problem. I'll explain some old and new reasons to suspect that there may be powerful emergent symmetries organizing the spectrum of large N confining theories. In a setting where the large N spectrum can be explored analytically, I'll show some recent concrete evidence for the existence of surprising emergent symmetries in the large N limit. These symmetries end up implying some remarkable relations between 4D confining theories and 2D conformal field theories, as well as, for instance, intricate Bose-Fermi correlations in manifestly non-supersymmetric large N theories.