  We study the effects of an $ab$-surface on the vortex-solid to vortex-liquid
transition in layered superconductors in the limit of vanishing inter-layer
Josephson coupling. We derive the interaction between pancake vortices in a
semi-infinite sample and adapt the density functional theory of freezing to
this system. We obtain an effective one-component order-parameter theory which
can be used to describe the effects of the surface on vortex-lattice melting.
Due to the absence of protecting layers in the neighbourhood of the surface,
the vortex lattice formed near the surface is more susceptible to thermal
fluctuations. Depending on the value of the magnetic field, we predict either a
continuous or a discontinuous surface melting transition. For intermediate
values of the magnetic field, the surface melts continuously, assisting the
formation of the liquid phase and suppressing hysteresis above the melting
transition, a prediction consistent with experimental results. For very low and
very high magnetic fields, the surface melts discontinuously. The two different
surface melting scenarios are separated by two surface multicritical points,
which we locate on the melting line.
