  We develop basic notions and methods of algebraic geometry over the algebraic
objects called hyperrings. Roughly speaking, hyperrings generalize rings in
such a way that an addition is `multi-valued'. This paper largely consisits of
two parts; algebraic aspects and geometric aspects of hyperrings. We first
investigate several technical algebraic properties of a hyperring. In the
second part, we begin by giving another interpretation of a tropical variety as
an algebraic set over the hyperfield which canonically arises from a totally
ordered semifield. Then we define a notion of an integral hyperring scheme
$(X,\mathcal{O}_X)$ and prove that $\Gamma(X,\mathcal{O}_X)\simeq R$ for any
integral affine hyperring scheme $X=Spec R$.
