The existence of solitary-wave solutions to the three-dimensional water-wave problem with strong surface-tension effects is predicted by the KP-I model equation (Kadomtsev and Petviashvili). The term solitary wave describes any solution which has a pulse-like profile in its direction of propagation, and the KP-I equation admits three types of solitary wave. A line solitary wave is spatially homogeneous in the direction transverse to its direction of propagation, while a periodically modulated solitary wave is periodic in the transverse direction. A fully localised solitary wave on the other hand decays to zero in all spatial directions. In this talk I outline mathematical results which confirm the existence of all three types of solitary wave for the full gravity-capillary water-wave problem in its usual formulation as a free-boundary problem for the Euler equations. The line solitary wave is found by establishing the existence of a two dimensional invariant manifold containing a homoclinic orbit (Kirchg¨assner). The periodically modulated solitary waves are created when the line solitary wave undergoes a dimension-breaking bifurcation in which it spontaneously loses its spatial homogeneity in the transverse direction; an infinite-dimensional version of the Lyapunov centre theorem is the main ingredient in the existence theorem (Groves, Haragus and Sun). The fully localised solitary wave is obtained by a variational argument based upon the mountain-pass lemma and the concentration-compactness theorem (Groves and Sun).