A **coordinate system** is a system for assigning an tuple of scalars to each point in an n-dimensional space. "Scalars" in many cases means real numbers, but, depending on context, can mean complex numbers or members of any of many other rings or ring-like algebraic structures.

Although any specific coordinate system is useful for numerical calculations in a given space, the space itself is independent of any particular choice of coordinates. Some choices of coordinate systems lead to paradoxes, for example, close to a black hole, but can be understood by changing the choice of coordinate system.

A **coordinate transformation** is a conversion from one system to another, to describe the same space.

An example of a coordinate system is to describe a point P in the Euclidean space **R**^{n} by an n-tuple P=(r_{1},...,r_{n}) of real numbers r_{1},...,r_{n}.

These numbers r_{1},...,r_{n} are called the *coordinates* of the point P.

If a subset S of an Euclidean space is mapped continuously onto another topological space, this defines coordinates in the image of S.

Some coordinate systems are the following:

- The Cartesian coordinate system (also called the "rectangular coordinate system"), which, for three-dimensional flat space, uses three numbers representing distances.
- For any finite-dimensional vector space and any basis, the coefficients of the basis vectors can be used as coordinates. Changing the basis is a coordinate transformation, a linear transformation that can be summarized by a matrix, and is computationally the same as a mapping of points to other points keeping the bases the same: e.g. in 2D:
- a clockwise rotation is a mapping of points to other points which changes the coordinates the same as keeping the points in place but rotating the coordinate axes anti-clockwise.
- an expansion by a factor two in the direction of one basis vector is a mapping of points to other points which changes the coordinates the same as keeping the points in place but halving the magnitude of that basis vector (in both cases the corresponding coordinate is doubled).
- a mapping of points to other points which distorts a rectangle to a parallellogram changes the coordinates the same as keeping the points in place but changing the basis vectors from being two sides of that parallellogram to perpendicular ones, two sides of that rectangle.

- The polar coordinate systems
- Cylindrical coordinate system represents a point in space by an angle, a distance from the origin and a height.
- Spherical coordinate system represents a point in space with two angles and a distance from the origin.

- Celestial coordinate system
- Horizontal coordinate system
- Equatorial coordinate system - based on Earth rotation
- Ecliptic coordinate system - based on Solar System rotation
- Galactic coordinate system - based on Milky Way rotation

- extragalactic coordinate systems
- supergalactic coordinate system - based on plane of local supercluster of galaxies
- comoving coordinates - valid to particle horizon

- Binary coordinate system