  The effects of gravitational lensing of the cosmic microwave background (CMB)
have been measured at high significance with existing data and will be measured
even more precisely in future surveys. Reversing the effects of lensing on the
observed CMB temperature and polarization maps provides a variety of benefits.
Delensed CMB spectra have sharper acoustic peaks and more prominent damping
tails, allowing for improved inferences of cosmological parameters that impact
those features. Delensing reduces $B$-mode power, aiding the search for
primordial gravitational waves and allowing for lower variance reconstruction
of lensing and other sources of secondary CMB anisotropies. Lensing-induced
power spectrum covariances are reduced by delensing, simplifying analyses and
improving constraints on primordial non-Gaussianities. Biases that result from
incorrectly modeling nonlinear and baryonic feedback effects on the lensing
power spectrum are mitigated by delensing. All of these benefits are possible
without any changes to experimental or survey design. We develop a
self-consistent, iterative, all-orders treatment of CMB delensing on the curved
sky and demonstrate the impact that delensing will have with future surveys.
