- The phase diagram of elemental liquids has been found to be surprisingly rich, including variations in the melting curve and transitions in the liquid phase. The effect of these transitions in the liquid state on the shape of the melting curve is analyzed. First-order phase transitions intersecting the melting curve imply piecewise continuous melting curves, with solid-solid transitions generating upward kinks or minima and liquid-liquid transitions generating downward kinks or maxima. For liquid-liquid phase transitions proposed for carbon, phosphorous selenium, and possibly nitrogen, we find that the melting curve exhibits a kink. Continuous transitions imply smooth extrema in the melting curve, the curvature of which is described by an exact thermodynamic relation. This expression indicates that a minimum in the melting curve requires the solid compressibility to be greater than that of the liquid, a very unusual situation. This relation is employed to predict the loci of smooth maxima at negative pressures for liquids with anomalous melting curves. The relation between the location of the melting curve maximum and the two-state model of continuous liquid-liquid transitions is discussed and illustrated by the case of tellurium.