This thesis is concerned with the development of new diagnostic tools in the field of molecular spectroscopy and spectropolarimetry for studying solar and stellar magnetism. Molecules are formed in a large variety of cool astronomical objects, ranging from comets in the solar system to galaxies at high redshifts, and exhibit high temperature, pressure, and magnetic field sensitivities. For these objects, molecular spectroscopy provides a unique tool to study their physical properties and in particular their magnetic fields. For this purpose, we establish the theoretical foundations for the analysis of molecules, which involves a general introduction to molecular spectroscopy with a particular focus on magnetic fields and polarization in a molecule. We then apply the theory to the molecules MgH, TiO, CaH, and in particular to FeH, for the study of a broad range of stellar objects, starting from our Sun and proceeding to G-K-M stars. We investigate the usability and the limitations of the various molecules as astrophysical magnetic field diagnostics, placing the main emphasis on FeH bands, which are the most sensitive indicators of magnetic fields in cool atmospheres. We present for the first time spectropolarimetric sunspot observations of the FeH F4Δ-X4Δ system. The diagnostic capabilities of this particular system for probing solar and stellar magnetic fields are investigated. It is shown that the current theory is able to successfully reproduce the magnetic properties of many lines of the FeH F4Δ-X4Δ system, but fails in the case of some highly magnetically sensitive lines. Therefore, we develop in the next step a new quantum-mechanical approach for the treatment of the Zeeman effect in the FeH F4Δ-X4Δ system, which takes into account the thus far disregarded effects of an unknown nearby perturbation. This semi-empirical model is based on the comparison of observed and calculated sunspot spectra and allows to finally use the FeH F4Δ-X4Δ system for inferring magnetic field strengths in different astrophysical objects, such as M dwarfs. On cool stars, atomic lines vanish among the dominating presence of molecular features, and thus, it is of advantage to utilize molecular lines instead. The study of M dwarfs is of special interest, because the transition from stars with an outer convection zone to fully convective stars takes place. In the latter, the solar type dynamo must be replaced by an alternative mechanism, such as a turbulent dynamo, to amplify magnetic fields. With the previous development of the FeH F4Δ-X4Δ system into a diagnostic tool, we determine magnetic fields on a sample of M stars through a comparison of synthetic and observed spectral lines, which are especially magnetically sensitive. Our study is then extended to the analysis of the usefulness of various molecules, FeH, MgH, TiO, and CaH, as diagnostics for exploring stellar magnetism on active G-K-M stars. We present the temperature range in which the selected molecules can serve as indicators for magnetic fields on highly active cool stars. In particular, the detection of circular polarization in MgH and FeH lines on M dwarfs is shown and supports the predictions of our modeling. In the next step, we utilize the successful direct modeling of MgH and TiO lines in sunspots as a starting point for inversions, i.e. the determination of the physical parameters of the sunspot atmosphere. Including both molecular and atomic lines in these preliminary inversions promises a dramatic improvement of the deduced atmosphere model. Molecular lines prove to be excellent diagnostics to provide information on solar and stellar magnetic fields. Advances in the constantly growing field of molecular spectroscopy and spectropolarimetry depend largely on the following criteria: On one hand, from a theoretical point of view, much work remains to be done for the mostly very complicated cases of the molecular Zeeman and Paschen-Back effects. On the other hand, improved observational instrumentation at the new generation of large telescopes, such as highly sensitive spectropolarimeters, are designed to measure with high precision the polarization state of the received photons and may reveal new polarization phenomena in molecules.