The most prominent features on the surface of the solar disc are the sunspots, which have been studied since their detection in the 17th century. Despite this years and centuries of scientific investigations, surprisingly many facets of sunspots are not well understood. In this thesis, the properties of a sunspots' penumbra are derived from the observed spectra by means of an inversion, a forward modeling technique. A method for the construction of a 3-dimensional model of the field topology from the inversion results is presented. It offers the possibility to build a toy-model of the spot. This allows to investigate the relationships between the geometry and other properties of the sunspot like flow velocities, intensity, temperature, or field strength. The application of the procedure to time series of observations will allow to study the temporal evolution of the geometry and the other properties, to determine the physical processes happening and their respective drivers. A second part of the thesis is devoted to the analysis of isolated field concentrations in the surroundings of the sunspot. These isolated magnetic elements are assumed to be the elementary building blocks of the solar surface magnetism. In the turbulent surroundings of the solar atmosphere, only relatively strong concentrations with a field strength above 1 kG are assumed to be stable at all. Such field concentrations produce brightenings in the G band, where a lot of spectral lines of the CH molecule are present. Inside magnetic fields, the density is reduced, as the magnetic pressure contributes to the pressure balance equation. Thus, the reduced density allows to see to deeper layers of the atmosphere with higher temperature. If the CH-molecules dissociate due to the higher temperature, the spectral lines disappear and the intensity increases strongly. Due to the small size of the field concentration, the brightenings are strongly localized (''G-band Bright Points''). In this study, we find a much larger variability of the Bright Point properties than expected, which puts their elementary nature in some doubt. In the conclusions of this work, the resulting 3-D topology of the sunspot is used in an attempt to develop a consistent picture of the development and the fine structure of sunspots.