In mathematics, a cusp is a point on a curve where two branches, coming from different directions, meet and have a common tangent. If the two branches of the curve approach the tangent from opposite sides the cusp is called a keratoid (from the Greek kera for horn) or first-order cusp. This is the case, for example, with the curve given by the equation y ² = x ²y + x ⁵. If the two branches of the curve approach the tangent from the same side the result is a ramphoid or second-order cusp. The curvature at a cusp is infinite.

"Cusp" derives from the Latin cuspis for sharp. Outside of mathematics, the points of a crescent moon are called cusps and the sharp pointed premolar teeth of children are known as bicuspids.