Rotation in mathematics is a concept originating in geometry. Any rotation is a motion of a certain space that preserves at least one point. It can describe, for example, the motion of a rigid body around a fixed point. Rotation can have a sign (as in the sign of an angle): a clockwise rotation is a negative magnitude so a counterclockwise turn has a positive magnitude. A rotation is different from other types of motions: translations, which have no fixed points, and (hyperplane) reflections, each of them having an entire (n − 1)-dimensional flat of fixed points in a n-dimensional space.

Mathematically, a rotation is a map. All rotations about a fixed point form a group under composition called the rotation group (of a particular space). But in mechanics and, more generally, in physics, this concept is frequently understood as a coordinate transformation (importantly, a transformation of an orthonormal basis), because for any motion of a body there is an inverse transformation which if applied to the frame of reference results in the body being at the same coordinates. For example, in two dimensions rotating a body clockwise about a point keeping the axes fixed is equivalent to rotating the axes counterclockwise about the same point while the body is kept fixed. These two types of rotation are called active and passive transformations.

The rotation group is a Lie group of rotations about a fixed point. This (common) fixed point or center is called the center of rotation and is usually identified with the origin. The rotation group is a point stabilizer in a broader group of (orientation-preserving) motions.

For a particular rotation:

A representation of rotations is a particular formalism, either algebraic or geometric, used to parametrize a rotation map. This meaning is somehow inverse to the meaning in the group theory.

Rotations of (affine) spaces of points and of respective vector spaces are not always clearly distinguished. The former are sometimes referred to as affine rotations (although the term is misleading), whereas the latter are vector rotations. See the article below for details.

A motion of a Euclidean space is the same as its isometry: it leaves the distance between any two points unchanged after the transformation. But a (proper) rotation also has to preserve the orientation structure. The "improper rotation" term refers to isometries that reverse (flip) the orientation. In the language of group theory the distinction is expressed as direct vs indirect isometries in the Euclidean group, where the former comprise the identity component. Any direct Euclidean motion can be represented as a composition of a rotation about the fixed point and a translation.

In one-dimensional space, there are only trivial rotations. In two dimensions, only a single angle is needed to specify a rotation about the origin – the angle of rotation that specifies an element of the circle group (also known as U(1)). The rotation is acting to rotate an object counterclockwise through an angle θ about the origin; see below for details. Composition of rotations sums their angles modulo 1 turn, which implies that all two-dimensional rotations about the same point commute. Rotations about different points, in general, do not commute. Any two-dimensional direct motion is either a translation or a rotation; see Euclidean plane isometry for details.