  The solar wind is the extension of the Sun's hot and ionized corona, and it
exists in a state of continuous expansion into interplanetary space. The radial
distance at which the wind's outflow speed exceeds the phase speed of Alfvenic
and fast-mode magnetohydrodynamic (MHD) waves is called the Alfven radius. In
one-dimensional models, this is a singular point beyond which most fluctuations
in the plasma and magnetic field cannot propagate back down to the Sun. In the
multi-dimensional solar wind, this point can occur at different distances along
an irregularly shaped "Alfven surface." In this article, we review the
properties of this surface and discuss its importance in models of solar-wind
acceleration, angular-momentum transport, MHD waves and turbulence, and the
geometry of magnetically closed coronal loops. We also review the results of
simulations and data analysis techniques that aim to determine the location of
the Alfven surface. Combined with recent perihelia of Parker Solar Probe, these
studies seem to indicate that the Alfven surface spends most of its time at
heliocentric distances between about 10 and 20 solar radii. It is becoming
apparent that this region of the heliosphere is sufficiently turbulent that
there often exist multiple (stochastic and time-dependent) crossings of the
Alfven surface along any radial ray. Thus, in many contexts, it is more useful
to make use of the concept of a topologically complex "Alfven zone" rather than
one closed surface. This article also reviews how the Polarimeter to Unify the
Corona and Heliosphere (PUNCH) mission will measure the properties of the
Alfven surface and provide key constraints on theories of solar-wind
acceleration.
