Existential least fixpoint logic (ELFP) is a logic with a least fixpoint operator but only existential quantification. It arises in many areas of computer science including logic programming, database theory, program verification, complexity theory, and recursion theory on abstract structures. A sequent calculus (Gentzen-style deductive system) for this logic is presented and proved to be complete. Basic model theoretic facts about ELFP are derived from the completeness theorem and the construction used in its proof. The relationship of these model theoretic facts to logic programming and database queries is explored.