Let $F$ be a field, and let $\Gamma$ be a totally ordered abelian group. A valuation is a group homomorphism $v : F \to \Gamma$ such that $v(x + y) \geq \min(v(x), v(y))$. On the other hand, a valuation ring $O \subseteq F$ is a ring such that for every $x \in F^\times$, we have $x \in O$ or $x^{-1} \in O$. It is classical that the valuations on a field (up to equivalence) are in bijective correspondence with the valuation rings of that field.
Valuation theory also has important applications in number theory (since valuations correspond to prime ideals in a number field), and to algebraic geometry (because they correspond to points in affine space, among other reasons).
My question is: what work has been done to generalize valuations? The definition of a valuation makes sense without any modification if $\Gamma$ is an abelian lattice-ordered group instead of an abelian ordered group, and there are plenty of integral domains people care about that are not valuation rings. Have people studied "valuation-like" structures where the definition of a valuation is tweaked or generalized in some way? Have people studied "valuation-like rings" where that definition is tweaked or generalized in some way? Are there established correspondences between generalized valuations and generalized valuation rings?