  Alfv\'en waves are a particular class of magnetohydrodynamic waves relevant
in many astrophysical and laboratory plasmas. In partially ionized plasmas the
dynamics of Alfv\'en waves is affected by the interaction between ionized and
neutral species. Here we study Alfv\'en waves in a partially ionized plasma
from the theoretical point of view using the two-fluid description. We consider
that the plasma is composed of an ion-electron fluid and a neutral fluid, which
interact by means of particle collisions. To keep our investigation as general
as possible we take the neutral-ion collision frequency and the ionization
degree as free parameters. First, we perform a normal mode analysis. We find
the modification due to neutral-ion collisions of the wave frequencies and
study the temporal and spatial attenuation of the waves. In addition, we
discuss the presence of cut-off values of the wavelength that constrain the
existence of oscillatory standing waves in weakly ionized plasmas. Later, we go
beyond the normal mode approach and solve the initial-value problem in order to
study the time-dependent evolution of the wave perturbations in the two fluids.
An application to Alfv\'en waves in the low solar atmospheric plasma is
performed and the implication of partial ionization for the energy flux is
discussed.
