This paper introduces a notion of existential continuation and provides a theoretical basis for continuation passing style (CPS) programming in logic programming. We present an algorithm, called CPS conversion, which converts a definite clause program into a CPS program, correspondingly to a predicate partition, an analogue of mode pattern specifying input and output for each predicate. Depending on the choice of a predicate partition, the resulting program can have various execution modes from bottom-up to top-down computations. We then give a proof, on the basis of a meaning preserving logic program transformation system, that the CPS conversion preserves the meaning of programs.