Viscous and persistent foams made from surfactant solutions were obtained by subjecting coarse foams to flow through two or three refining screens. A screen refines the incoming coarse foam by shearing and/or extending and finally breaking up the bubbles as they approach and flow through the screen openings. For a screen to be effective, the generated strain rate, residence time and draw ratio should exceed the respective threshold values for bubble breakup. It is shown that all these threshold values can be lumped into a single parameter, that of the foam velocity through the screen openings. This velocity incorporates the characteristics of the screen, incoming foam, and the approaching foam velocity. For the resulting foam to be viscous and persistent, its velocity through the screen should be in a range, whose upper and lower limits depend on the viscosity of the solution and the characteristics of the incoming foam.