Atomistic mechanism of field emission is discussed in relation to the concept of local work function, which plays an important role for understanding materials properties of nano-structures on surfaces. First, the concept of local work function is clarified on the basis of the nano-scale potential distribution over the fringe of surface. Next, present status is described on the first-principles theory of the work function and field emission properties on metal surfaces. Further, various fundamental issues of field emission still to be resolved are discussed. The topics include various many body effects on the field emission, such as emission from superconductors or strongly correlated materials as well as from nano-particles weakly coupled with the surface. Finally recent developments of approaches probing local work function by SPM are reviewed and further problems are introduced.