We introduce a general theory of quantitative and metric rewriting systems, namely systems with a rewriting
relation enriched over quantales modelling abstract quantities. We develop theories of abstract and term-based
systems, refining cornerstone results of rewriting theory (such as Newman’s Lemma, Church-Rosser Theorem,
and critical pair-like lemmas) to a metric and quantitative setting. To avoid distance trivialisation and lack of
confluence issues, we introduce non-expansive, linear term rewriting systems, and then generalise the latter
to the novel class of graded term rewriting systems. These systems make quantitative rewriting modal and
context-sensitive, this way endowing rewriting with coeffectful behaviours.