Local error bounds play a fundamental role in mathematical programming and variational analysis. They are used e.g. as constraint qualifications in optimization, in developing calculus rules for generalized derivatives in nonsmooth and set-valued analysis, and they serve as a key ingredient in the design and convergence analysis of Newton-type methods for solving systems of possibly nonsmooth constrained equations with possibly nonisolated solutions. In this paper, we derive natural relations between several error bound properties that have attracted interest in rather different areas. More precisely, we establish equivalences between Lipschitzian error bound properties on the one hand, and the subtransversality of certain sets, and the metric subregularity of certain set-valued mappings, on the other hand. As a consequence, sufficient conditions developed with respect to one of these properties can be used to guarantee any of the others as well. Exemplarily, we will use Mordukhovich’s normal qualification condition as the natural sufficient condition for the equivalent properties just mentioned. Particular attention will be paid to Lipschitzian error bounds for smooth systems of constrained equations, and nondifferentiable composite equations, and the obtained results will be applied to guarantee an error bound for a complementarity system over a closed convex cone.