In this chapter, we introduce Gromov-Witten invariant, quantum cohomology, GromovWitten potential, and Floer cohomology on symplectic manifolds, and in connection with these, we describe Gromov-Witten type invariant, quantum type cohomology, Gromov-Witten type potential and Floer type cohomology on almost contact metric manifolds. On the product of a symplectic manifold and an almost contact metric manifold, we induce some relations between Gromov-Witten type invariant and quantum cohomology and quantum type invariant. We show that the quantum type cohomology is isomorphic to the Floer type cohomology.