Hypergroups are objects like groups but with addition taking possibly many values. Likewise, hyperrings and hyperfields are objects similar to rings and fields, but with multivalued addition. Hyperfields provide a convenient tool in axiomatizing the algebraic theory of quadratic forms and in this talk we shall focus on three such applications. Firstly, we shall show how Witt equivalence of fields can be conveniently expressed in the language of hyperfields and will present some recent results on Witt equivalence of function fields over global and local fields. Secondly, we shall show how orderings of higher level can be defined for hyperrings and hyperfields, and, consequently, how they can be used to provide an axiomatic framework to study forms of higher order. Finally, we shall define the category of, so called, presentable fields and define their Witt rings, thus providing yet another machinery to study quadratic forms over fields. The results presented in this talk were obtained jointly with Murray Marshall and Krzysztof Worytkiewicz.