We study here the properties of a family of monoids, which we call the rational monoids, and which are monoids with a multiplication of low complexity. A monoid is rational if its multiplication may be described by a rational function from a free monoid into itself. The main results are that rational monoids, like finite ones, have the properties that Green's relations and are equal and that Kleene's theorem holds in rational monoids, as in free ones. Every monoid described so far, in which Kleene's theorem holds, is a rational monoid. The closure of the family of rational monoids under Rees' quotient, direct product, and free product is then studied. Extensions of rational monoids will be considered in a forthcoming paper.