Experimental measurements of the advancing head of a gravity current are described. The head, of depth d in mean cross-section and advancing at mean speed U, has an elevated nose whose mean height h satisfies h/d = 0·61 Re-0·23±0·01 through the range 300 < Re < 10000 (Re = Ud/ν). A shifting pattern of lobes and clefts ranges across the head, in which the mean lobe width $\overline{b}$ is found experimentally to satisfy $\overline{b}/d = 7.4\,Re^{-0.39\pm 0.02}$. By moving the floor in the direction of the current, or by introducing a thin dense layer beneath it, the lobes and clefts can be suppressed and a purely two-dimensional flow obtained. It is concluded that the lower boundary plays an essential role in determining the substructure of the head and that this originates in convective instability as the head rides over less dense fluid.