In this talk I will present a survey of some of the famous results and examples in the classical theory of minimal and constant mean curvature surfaces in R^3. The first examples of minimal surfaces were found by Euler (catenoid) around 1741, Muesner (helicoid) around 1746 and Riemann (Riemann minimal examples) around 1860. The classical examples of non-zero constant mean curvature surfaces are the Delaunay surfaces of revolution found in 1841, which include round spheres and cylinders. My talk is full of beautiful computer graphics pictures of these and other classical surfaces, which hopefully will delight the audience. My lecture will also cover the classical contributions to the calculus of variations by Plateau, Lagrange, Schwarz and Lord Kelvin. At the end of my talk i will explain the final classification result of Meeks-Tinaglia in 2016 that the only complete simply-connected surfaces embedded in R^3 of constant mean curvature are the plane, the helicoid and round spheres (this depends on previous work of Colding-Minizcozzi and Meeks -Rosenberg), and the Meeks-Perez-Ros theorem that the only non-planar properly embedded minimal surfaces in R^3 of genus 0 are the classical ones found by Euler and Riemann. Given time I will mention the recent Meeks-Tinaglia theorem that the only complete annuli embedded in R^3 of constant mean curvature are Euler's catenoid and the famous revolution surfaces of Delaunay. Finally I will state the recent classification theorem of Meeks-Mira-Perez-Ros for constant mean curvature spheres in any homogeneous 3-manifold.