This thesis deals with the two topics "Thermodynamic equilibrium studies in the TiAl system" and "Thermodynamic non-equilibrium studies in the TiAl system". In both subject areas a systematic comparison of the results obtained from the experiment and the corresponding mathematical simulation is carried out. The first subject area contains a comprehensive mathematical representation of the Calculation of Phase Diagrams (CALPHAD) method, which is based on the principle of minimizing the total Gibbs energy function of a thermodynamic system. By means of the Lagrangian multiplier method, a Lagrangian function of the overall system is constructed, which is minimized under certain constraints. This makes it possible to calculate phase equilibria (phase diagrams and phase fraction diagrams) in technically relevant multicomponent systems. For the practical application of this generally valid methodology, an experimental determination of phase fraction diagrams in the specification window of the TNM alloy is carried out by means of ex-situ (X-ray diffraction) and in-situ tests (high-energy X-ray diffraction). For the thermodynamic description of phase fraction diagrams of the intermetallic TNM alloy, which was developed at the Montanuniversität Leoben, a specially optimized thermodynamic database is constructed for this alloy. The comparison between experiment and simulation shows very clearly that the experimental phase fraction diagrams can be described by the newly optimized TNM assessment in very wide temperature ranges. Within the second topic, the isothermal hot forming behavior of the TNM alloy is investigated. The isothermal deformation behavior of the TNM alloy is systematically studied in the three-phase field region by means of laboratory experiments (Gleeble experiments) and experimental flow curves are recorded. The deformed microstructure is investigated by means of metallographic characterization methods (Scanning Electron Microscopy and Electron Backscatter Diffraction), whereby the dynamically recrystallized grain sizes and the dynamically recrystallized volume fraction are determined. Starting from a mathematical description of the experimental flow curves (via phenomenologically and physically based constitutive models), a correlation between experiment and simulation is performed as far as the dynamically recrystallized structural parameters are concerned. To study the microstructural damage of the isothermally deformed TNM alloy, processing maps are calculated according to the Dynamic Materials Model (DMM) and the results are compared with the deformed microstructure.