ABSTRACT
This chapter introduces the basic notions of the theory of categories using natural examples directly related to the theory of graphs. A category is the natural algebraic notion that captures the associativity property of composition of maps. The chapter emphasises that the way of thinking in the realm of categories is not only natural but both simple and powerful as it provides a new insight into some basic manipulations with such simple mathematical objects as graphs and relations. The language of categories is extremely powerful as it allows us to encompass many situations often found in Mathematics. Thus, it is clear that categories wherein all morphisms are isomorphisms are called to play an important role. Although it may not be a common opinion, one of the main applications in physics of groupoid theory could be to put together internal and external symmetries of physical systems. A common ground for groupoids in Physics and Mathematics is their application in noncommutative geometry.
