We study the notion of computable categoricity of computable structures, comparing it especially to the notion of relative computable categoricity and its relativizations. We show that every 1-decidable computably categorical structure is relatively Δ02-categorical. We study the complexity of various index sets associated with computable categoricity and relative computable categoricity. We also introduce and study a variation of relative computable categoricity, comparing it to both computable categoricity and relative computable categoricity and its relativizations.