An enumerative invariant theory in algebraic geometry, differential geometry, or representation theory, is the study of invariants whichcount'-(semi) stable objects with fixed topological invariants in some geometric problem, by means of a virtual class in some homology theory for the moduli spaces of -(semi) stable objects. Examples include Mochizuki's invariants counting coherent sheaves on surfaces, Donaldson-Thomas type invariants counting coherent sheaves on Calabi-Yau 3-and 4-folds and Fano 3-folds, and Donaldson invariants of 4-manifolds. We make conjectures on new universal structures common to many enumerative invariant theories. Any such theory has two moduli spaces , , where the second author (see https://people. maths. ox. ac. uk/~ joyce/hall. pdf) gives the structure of a graded vertex algebra, and a graded Lie algebra, closely related to . The virtual classes take values in . In most such theories, defining when (in gauge theory, when the moduli space contains reducibles) is a difficult problem. We conjecture that there is a natural way to define invariants in homology over , with when , and that these invariants satisfy a universal wall-crossing formula under change of stability condition , written using the Lie bracket on . We prove our conjectures for moduli spaces of representations of quivers without oriented cycles. Versions of our conjectures in algebraic geometry using Behrend-Fantechi virtual classes are proved in the sequel [arXiv: 2111.04694].