In this talk, we investigate the local differential geometry of wave fronts and frontals. Wave fronts and frontals are surfaces in Euclidean 3-space and may admit singularities. Normal forms are widely adopted to explore the local differential geometry of singular surfaces. On the other hand, a framed surface is a smooth surface in Euclidean 3-space with a moving frame. Using the moving frame, the basic invariants and curvatures of framed surfaces are introduced. Combining the theories of the normal forms and the framed surfaces, we derive not only the normal forms for the infinite-type case of real analytic flat cuspidal edges and swallowtails, but also those for the finite-type case. Moreover, a real analytic cuspidal cross cap is never a flat surface.