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In this thesis, the Dyson-Schwinger - Bethe-Salpeter formalism isinvestigated and used to study the meson spectrum at zero temperature, as wellas the chiral phase transition in finite-temperature QCD. First, theapplication of sophisticated matrix algorithms to the numerical solution ofboth the homogeneous Bethe-Salpeter equation (BSE) and the inhomogeneous vertexBSE is discussed, and the advantages of these methods are described in detail.Turning to the finite temperature formalism, the rainbow-truncated quarkDyson-Schwinger equation is used to investigate the impact of different formsof the effective interaction on the chiral transition temperature. A strongmodel dependence and no overall correlation of the value of the transitiontemperature to the strength of the interaction is found. Within one model,however, such a correlation exists and follows an expected pattern. In thecontext of the BSE at zero temperature, a representation of the inhomogeneousvertex BSE and the quark-antiquark propagator in terms of eigenvalues andeigenvectors of the homogeneous BSE is given. Using the rainbow-laddertruncation, this allows to establish a connection between the bound-state polesin the quark-antiquark propagator and the behavior of eigenvalues of thehomogeneous BSE, leading to a new extrapolation technique for meson masses.This is used to study the ground- and excited-state meson spectrum for allquark masses from light to bottom, for pseudoscalar, scalar, vector,axialvector and tensor mesons. Good agreement with experiment is found, e.g.,for all ground states in the bottomonium system. In addition, new applicationsof the inhomogeneous vertex BSE, such as the possibility to calculate on-shellquantities like decay constants, are investigated. Finally, we study theinfluence of the infrared behavior of the effective interaction on propertiesof pi and rho mesons.

Properties of quarks and mesons in the Dyson-Schwinger/Bethe-Salpeter approach