In geometry, a star polygon is a type of non-convex polygon. Regular star polygons have been studied in depth; while star polygons in general appear not to have been formally defined, certain notable ones can arise through truncation operations on regular simple or star polygons.

Branko Grünbaum identified two primary usages of this terminology by Johannes Kepler, one corresponding to the regular star polygons with intersecting edges that do not generate new vertices, and the other one to the isotoxal concave simple polygons.

Polygrams include polygons like the pentagram, but also compound figures like the hexagram.

One definition of a star polygon, used in turtle graphics, is a polygon having q ≥ 2 turns (q is called the turning number or density), like in spirolaterals.

Star polygon names combine a numeral prefix, such as penta-, with the Greek suffix -gram (in this case generating the word pentagram). The prefix is normally a Greek cardinal, but synonyms using other prefixes exist. For example, a nine-pointed polygon or enneagram is also known as a nonagram, using the ordinal nona from Latin.^{[citation needed]} The -gram suffix derives from γραμμή (grammḗ), meaning a line. The name star polygon reflects the resemblance of these shapes to the diffraction spikes of real stars.

A regular star polygon is a self-intersecting, equilateral, and equiangular polygon.

A regular star polygon is denoted by its Schläfli symbol {p/q}, where p (the number of vertices) and q (the density) are relatively prime (they share no factors) and where q ≥ 2. The density of a polygon can also be called its turning number: the sum of the turn angles of all the vertices, divided by 360°.

The symmetry group of {p/q} is the dihedral group D_p, of order 2p, independent of q.