Scanning tunneling microscopy and related local probe methods have led to a novel perception of nanometer‐ and atomic‐scale structures and processes. Since the information is obtained directly in real space, the scanning probe techniques offer significant advantages for the investigation of nonperiodic structures at solid surfaces compared with diffraction techniques. Additionally, local probe methods allow the study of a large variety of physical properties of nanometer‐scale structures down to atomic resolution and even became useful for the fabrication of artificial nanometer‐scale structures. On the other hand, a large number of unresolved scientific and technological issues still remains. In this review, the focus is on two basic questions: what actually has been achieved and which fundamental physics issues need to be addressed further. The discussion will be restricted to topics in solid state physics.