An approach for computing scanning tunneling microscopy from first principles is proposed. Within the framework of Landauer-Büttiker theory, the conductance of a scanning tunneling microscope (STM) is obtained in terms of real-space Green functions, thereby taking into account incoherent tunneling processes and the tip-sample interaction but avoiding repeated STM tips as in a supercell approach. The approach is formulated within multiple-scattering theory, especially in the Korringa-Kohn-Rostoker method, but can be implemented in any Green-function method for electronic-structure calculations. Extensive tests are presented for planar and STM tunnel junctions involving Au(111) electrodes.