This paper considers from a simple physical point of view the Mössbauer effect, i.e., the "recoilless emission" of gamma rays from a nucleus bound in a crystal lattice. It begins with a discussion of the kinematics of gamma-ray emission from such a nucleus. The idealized case of a massive "lattice" characterized by a single frequency and the more realistic one and three-dimensional models are treated. We point up the fact that in the Mössbauer effect the lattice as a whole (the lattice center of mass) always recoils after photon emission, so that the term "recoilless emission" is in one sense misleading. We emphasize that the essence of the Mössbauer effect is not photon emission without recoil, but rather is photon emission without transfer of energy to internal degrees of freedom of the lattice. Using the basic ideas of quantum mechanics, namely, the rules for the manipulation of probability amplitudes (the so-called "transformation theory"), we calculate the probability for recoil without excitation of internal degrees of freedom, i.e., the Mössbauer f factor, on the assumption that the individual photon emissions, and consequent lattice recoil, are instantaneous. In Appendix A we discuss this question of instantaneous emission in some detail, and show how it is not in contradiction with the fact that the nuclear transition that leads to the gamma-ray emission has a finite half-width. In Appendix B those rules of transformation theory that are used in the body of the paper are summarized.