Discretization of curves is an ancient topic. However, there is no general theory or methodology in the literature treating this subject and even the definitions of basic concepts such as discrete curvature \(\kappa,\) discrete torsion \(\tau,\) and discrete Frenet frame are conflicting. In the present survey article the authors proceed to build three theories of discrete curves, all of which culminate in a discrete version of the Frenet equations. The approach is new and it is shown that the definitions of discrete length \(l,\) curvature \(\kappa\) and torsion \(\tau\) allow, given some \(l,\) \(\kappa,\) \(\tau,\) to recover a unique, up to rigid motion, discrete curve. To each of these three cases referred to as, inscribed, circumscribed and centered, there corresponds a natural differential-geometric way of defining the discretization of a smooth curve. The converse problem is also studied, that is, given a discrete curve, what is the natural differential-geometric way to spline the curve. In the final section the discrete surface theory is discussed.