Digital geometry is a discipline dealing with geometric properties of digital objects (also called digital pictures). These are usually modeled as sets of points with integer coordinates representing the pixels/voxels of the considered objects. Digital geometry is developed with the expectation that it would provide an adequate mathematical background for new advanced approaches and algorithms for various problems arising in image analysis and processing, computer graphics and related areas of visual computing. The present work provides a brief discussion on the motivation, basic directions, and achievements of digital geometry. A couple of typical examples of research problems and their solutions are also considered.