Advanced X-ray spectroscopic methods provide unique and critical data to study matter under extreme environmental conditions induced by high-intensity and high-energy lasers. The aim of this paper is to contribute to a contemporary discussion of the role of X-ray spectroscopy in the investigation of radiative properties of strongly coupled, highly correlated, and frequently weakly emissive plasma systems formed in matter irradiated by sub-petawatt and petawatt class lasers. After reviewing the properties of different X-ray crystal spectrometers, high-resolution X-ray diagnostic methods are surveyed with respect to their potential to study plasma-induced and externally induced radiation fields, suprathermal electrons, and strong electromagnetic field effects. Atomic physics in dense plasmas is reviewed with emphasis on non-Maxwellian non-LTE atomic kinetics, quasi-stationary and highly-transient conditions, hollow ion X-ray emission, and field-perturbed atoms and ions. Finally, we discuss the role of X-ray free electron lasers with respect to supplementary investigations of matter under extreme conditions via the use of controlled high-intensity radiation fields.