The main result of the paper is an intrinsic characterization of algebraizability in terms of the Leibniz operator [capital Greek] Omega, which associates with each theory [italic] T of a given deductive system [script] S a congruence relation [capital Greek] Omega [italic] T on the formula algebra.[capital Greek] Omega [italic] T identifies all formulas that cannot be distinguished from one another, on the basis of [italic] T, by any property expressible in the language of [script] S. The characterization theorem states that a deductive system [script] S is algebraizable if and only if [capital Greek] Omega is one-to-one and order-preserving on the lattice of [script] S-theories, and in addition preserves directed unions. Several other characteristics are given. The results and concepts are illustrated by a large number of examples from modal and intuitionistic logic, relevance logic, and classical predicate logic.