A cusp is a sharp point or apex, such as occurs in two dimensions at the end of a crescent, or in three dimensions at the tip of a cone or horn. There are a number of technical terms derived from it.

In mathematics, a cusp might mean

• a singular point of a curve such as that seen near (0,0) for

y = x^2/3;

• a conical singular point seen in hyperbolic geometry;
• a missing point on a modular curve that has to be supplied by compactification.

The term cusp form in modular form theory refers to the third of these meanings, implying a form that vanishes at cusps. There is an intricate theory of the connection of cuspidal phenomena with parabolic subgroups.

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In a separate field, cusp is a term in astrology for the boundary day(s) between zodiac signs.