Although hundreds of years old, printing processes offer great promise for efficient, environmentally friendly large-scale manufacture of flexible electronics and other nanoscale and microscale devices. The principal commercial challenge is to maximize throughput while minimizing defects. Meeting this commercial challenge will require addressing a number of basic scientific challenges concerning fluid flow and interfacial phenomena. Free-surface dynamics, contact-line motion, interfacial tension, wetting, electrohydrodynamics, and non-Newtonian rheology all play central roles in the successful operation of various printing processes. In his nearly 40-year career, Bud Homsy has made pioneering fundamental contributions to all of these areas, and his work exemplifies the elegant and careful investigation of complex phenomena. This talk will begin with an overview of printing processes and the associated fluid mechanics. Then, results from some recent boundary integral and finite-element simulations of surface-tension-driven liquid emptying from tiny cavities will be presented. The results help provide insight into gravure printing, a process in which liquid is removed at high speeds (~ m/s) from sub-millimeter-scale depressions engraved into a rotating cylinder.