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In this paper, we introduce the category of blueprints, which is a categoryof algebraic objects that include both commutative (semi)rings and commutativemonoids. This generalization allows a simultaneous treatment of ideals resp.\congruences for rings and monoids and leads to a common scheme theory. Inparticular, it bridges the gap between usual schemes and $\mathbb{F}_1$-schemes(after Kato, Deitmar and Connes-Consani). Beside this unification, the categoryof blueprints contains new interesting objects as "improved" cyclotomic fieldextensions $\mathbb{F}_{1^n}$ of $\mathbb{F}_1$ and "archimedean valuationrings". It also yields a notion of semiring schemes.  This first paper lays the foundation for subsequent projects, which aredevoted to the following problems: Tits' idea of Chevalley groups over$\mathbb{F}_1$, congruence schemes, sheaf cohomology, $K$-theory and a unifiedview on analytic geometry over $\mathbb{F}_1$, adic spaces (after Huber),analytic spaces (after Berkovich) and tropical geometry.

The geometry of blueprints. Part I: Algebraic background and scheme theory