We consider chemical reaction networks (CRN) with monomial reaction rates. For these models, we introduce and discuss the concepts of exact and approximate conservation laws, that are first integrals of the full and truncated sets of ODEs. For fast-slow CRNs, truncated ODEs describe the fast dynamics. We define compatibility classes as subsets of the species concentration space, obtained by equating the conservation laws to constants. A set of conservation laws is complete when the corresponding compatibility classes contain a finite number of steady states (quasi-steady states for approximate conservation laws of fast-slow systems). Complete sets of conservation laws can be used for reducing CRNs by variable pooling. We provide algorithmic methods for computing linear, monomial and polynomial conservation laws of CRNs and for testing their completeness. The resulting conservation laws and their completeness are are either independent or dependent on the parameters. In the latter case, we provide parametric case distinctions. In particular, we propose a new method to compute polynomial conservation laws by comprehensive Gr\"obner systems and syzygies. Keywords: First integrals, chemical reaction networks, polynomial conservation laws, syzygies, comprehensive Gr\"obner systems.