A chromatic geometry, or rainbow, is defined as follows. Let X and I be sets and \(\Delta\) : \(X\times X\to I\) a surjective function. For \(i\in I\) we write \(\Delta_ i\) for the inverse image in \(X\times X\) of i under \(\Delta\) and define \(\Delta \quad p_ i=\{(y,x): (x,y)\in \Delta_ i\}.\) A triple (X,\(\Delta\),I) is a chromatic geometry or rainbow if (1) there exists \(i_ 0\in I\) such that \(\Delta_{i_ 0}=\{(x,x): x\in X\},\) (2) for each \(i\in I\) there exists i \(p\in I\) such that \(\Delta\) \(p_ i=\Delta_{i\quad p}.\) The category of rainbows provides a geometric setting for studying permutation groups. In this paper the author presents some of the main properties of rainbows. In a later paper he will use the theory of rainbows to study a mathematical model of neural networks due to \textit{J. J. Hopfield} [Proc. Natl. Acad. Sci. USA 79, 2554-2558 (1982)].