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Vaughan Jones discovered unexpected connections between Richard Thompson's group and subfactor theory while attempting to construct conformal field theories (in short CFT). Among other this founded Jones' technology: a powerful new method for constructing actions of fraction groups which had numerous applications in mathematical physics, operator algebras, group theory and more surprisingly in knot theory and noncommutative probability theory. We propose and outline a program in the vein of Jones' work but where the Thompson group is replaced by a family of groups that we name forest-skein groups. These groups are constructed from diagrammatic categories, are tailor-made for using Jones' technology, capture key aspects of the Thompson group, and aim to better connect subfactors with CFT. Our program strengthens Jones' visionary work and moreover produces a plethora of concrete groups which satisfy exceptional properties. In this first article we introduce the general theory of forest-skein groups, provide criteria of existence, give explicit presentations, prove that their first L$^2$-Betti number vanishes, construct a canonical action on a totally ordered set, establish a topological finiteness theorem showing that many of our groups are of type $F_\infty$, and finish by studying a beautiful class of explicit examples.