Despite the long history of studies of active galactic nuclei (AGN), details of the structure of the accretion flow onto supermassive black holes are far from clear. Work presented in this thesis is directed at unveiling properties of AGN structure through broadband X-ray spectroscopy, with particular emphasis on the hard X-ray band (photon energies above 10 keV). With its unprecedented sensitivity in this energy band, the NuSTAR telescope provides the key observational diagnostics of the properties of the AGN X-ray source, the corona, and the surrounding gas in the accretion disk, the broad-line region, and the torus. The first study presented in this thesis focuses on measurements of the optical depth and the temperature of the plasma in the corona of an obscured AGN. Fitting theoretical spectral models for coronal emission to the NuSTAR data constrained these two basic physical parameters under the assumption of either spherical or disk-like geometry for the corona. The remainder of the thesis is dedicated to studies of the anisotropic obscuring structure broadly referred to as the torus. One of them is a case study of three heavily obscured AGN with spectra dominated by the X-ray light scattered and reprocessed in the torus, where it is possible to constrain one of the basic torus properties - its globally averaged column density. The following study presents the calculation of a new spectral model for reprocessing of the intrinsic X-ray continuum within the torus. Its added flexibility compared to previously available models allows for both the average column density of the torus and its covering factor to be constrained from broadband X-ray spectra of a wide variety of AGN. The final part of the thesis in based on a large survey of the local obscured AGN population performed with NuSTAR. Spectral modeling of more than a hundred individual AGN, including both old and new spectral models, is presented. From analyses of the X-ray data for a large and representative AGN sample, for the first time, it is found that their tori preferentially have high covering factors and average column densities close to unity optical depth for Compton scattering.