We report experiments on the shape and motion of millimetre-sized drops sliding down a plane in a situation of partial wetting. When the Bond number based on the component of gravity parallel to the plane $\Bo_{\alpha}$ exceeds a threshold, the drops start moving at a steady velocity which increases linearly with $\Bo_{\alpha}$. When this velocity is increased by tilting the plate, the drops change their aspect ratio: they become longer and thinner, but maintain a constant, millimetre-scale height. As their aspect ratio changes, a threshold is reached at which the drops are no longer rounded but develop a ‘corner’ at their rear: the contact line breaks into two straight segments meeting at a singular point or at least in a region of high contact line curvature. This structure then evolves such that the velocity normal to the contact line remains equal to the critical value at which the corner appears, i.e. to a maximal speed of dewetting. At even higher velocities new shape changes occur in which the corner changes into a ‘cusp’, and later a tail breaks into smaller drops (pearling transition). Accurate visualizations show four main results. (i) The corner appears when a critical non-zero value of the receding contact angle is reached. (ii) The interface then has a conical structure in the corner regime, the in-plane and out-of-plane angles obeying a simple relationship dictated by a lubrication analysis. (iii) The corner tip has a finite non-zero radius of curvature at the transition to a corner, and its curvature diverges at a finite capillary number, just before the cusp appears. (iv) The cusp transition occurs when the corner opening in-plane half-angle reaches a critical value of about 45°.