As is natural for all properties determined by surfaces and interfaces rather than by bulk, adhesion and friction increase in importance when the objects that come into contact  and move relative to one another are nanosized. In this talk I propose to review some theory and simulation work recently done in our group on topics and models that are inspired by nanofriction. Among them are: * Some strange phenomena determined by the sliding of periodic but incommensurate systems. * The effect of high temperatures and surface melting on the friction felt by an AFM tip. * The negative differential friction predicted for high-speed sliding coaxial nanotubes. * The diffusive-ballistic crossover predicted for the frictional slip time of gold nanoclusters on graphite.