The Bargmann transform is a unitary operator from L²(R) onto the Fock space F² of Gaussian integrable entire functions on the complex plane. Under the Bargmann transform, many classical operators on L²(R) take fascinating new forms on the Fock space. The talk will begin with several examples of operators on L²(R), including the Fourier transform, the Hilbert transform, annihilation operators, and creation operators. Motivated by the Hilbert transform, we will then define singular integral operators on the Fock space, examine several examples, formulate a conjecture about the boundedness of singular integral operators on F² , and present recent progress on the conjecture.