A discussion of the origins of ornamental art - illustrated by the oldest examples, dating mostly from the paleolithic and neolithic ages, and considered from the theory-of-symmetry point of view. Because of its multidisciplinary nature, it should interest a wide range of readers, including mathematicians, artists, art historians, architects, psychologists and anthropologists. The work represents the complete analysis of plane symmetry structures, so it can be used by artists as a guide to the creation of new symmetry patterns. Some parts of the contents (such as Chapter 4, about conformal symmetry, and Chapter 6, about modularity in art) give the reader an opportunity to develop computer programmes for producing images illustrating the corresponding symmetry forms.