•We discuss conceptual and technical limitations of ABTS+•, DPPH, and ORAC assays.•Steric accessibility to ABTS+• and DPPH radicals controls antioxidant reactions.•Kinetics is more relevant than stoichiometry for all antioxidant efficacy assays.•Kinetic patterns plus solvent/pH effects on rates distinguish HAT from SET mechanisms.•Redirection of assays to distinguish radical quenching mechanisms is recommended. Assays developed to measure radical scavenging ability of natural compounds have been used as a basis for ranking and recommending best foods for consumption. However, assays often were adapted for screening assays with inadequate consideration of reaction chemistry, particularly kinetics. Recent research results raise serious questions about the chemistry, execution, and application of these assays. This paper critically evaluates conceptual and technical issues that limit use and compromise validity of three commonly-used assays – TEAC/ABTS•+, DPPH, and ORAC. Recommendations are made for discontinuing use of ABTS•+ and DPPH radicals for measuring radical quenching, redirecting them instead to distinguishing electron transfer reaction mechanisms. Conditions required for accurate results in ORAC are reviewed, and recommendations are made for redirecting this assay to distinguishing compounds that quench radicals by hydrogen atom transfer. The mechanistic information so gained can be then applied to understanding how natural antioxidants can be used most effectively in foods.