<p>We study the properties of several boundaries of a proper geodesic metric space. We prove that the horofunction boundary is the weakest boundary for the continuous extension of the Gromov product, that is, a boundary admits a continuous extension of the Gromov product if and only if this boundary is finer than the horofunction boundary. As a corollary, for a Gromov hyperbolic metric space, the Gromov boundary is equivalent to the horofunction boundary if and only if the Gromov product has a continuous extension to the Gromov boundary. Moreover, by defining appropriate equivalence relations on the horofunction boundary, we construct two new boundaries for a proper geodesic metric space, which are both generalizations of the Gromov boundary for a Gromov hyperbolic metric space, and thus are called the generalized Gromov boundaries. We apply our results to the Teichmüller space. We give a new proof of the following result: the horofunction boundary of the Teichmüller space (with respect to the Teichmüller metric) is equivalent to the Gardiner–Masur boundary. And we characterize the two generalized Gromov boundaries of the Teichmüller space by two different quotient spaces of the space of projective measured foliations.</p>