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The fine structure of excitons is studied by magnetophotoluminescence spectroscopy of single self-assembled In (Ga) As/(Al) GaAs quantum dots. Both strength and orientation of the magnetic field are varied. In a combination with a detailed theoretical analysis, these studies allow us to develop a comprehensive picture of the exciton fine structure. Symmetry of the dot structures as well as its breaking cause characteristic features in the optical spectra, which are determined by the electron-hole exchange and the Zeeman interaction of the carriers. The symmetry breaking is either inherent to the dot due to geometry asymmetries, or it can be obtained by applying a magnetic field with an orientation different from the dot symmetry axis. From data on spin splitting and on polarization of the emission we can identify neutral as well as charged exciton complexes. For dots with weakly broken symmetry, the angular …