Abstract This thesis focuses on observational study of a variety of phenomena that occur in the vicinity of two types of compact stars: neutron stars (NS) and black holes (BH). These two compact star types are characterized by their enormous densities. The densities in NS cores exceed the normal nuclear density in heavy atomic nuclei. Therefore, by studying neutron stars, we can infer what happens to atomic nuclei when they are compressed. BHs are the densest stars in the universe: in fact, they are so dense that even light cannot escape from their surfaces. The entire mass of a BH is concentrated into an infinitely dense point called a singularity, which is shrouded from view by the event horizon. NSs and BHs display a large variety of observational phenomena when they are members in interactive binary star systems. In these binary systems, the compact star accretes matter from a companion star and a fraction of the available gravitational potential energy of the in-falling gas is converted into X-ray light. In this thesis, I have studied two different types of compact star systems using data from satellite instruments that can detect these X-rays. I have studied a class of accreting NSs called Accreting Millisecond Pulsars (AMP) and luminous extragalactic systems called Ultra-Luminous X-ray sources (ULX) that are thought to be powered by accretion onto BHs. In this thesis, I will demonstrate how we can infer physical properties of compact stars through X-ray spectroscopy and timing analysis, and I will also show how we have applied these methods to study the properties of AMPs and ULXs. The results we have obtained improve our understanding of these systems. We have discovered that spectra of AMPs vary strongly below the energy band that is commonly used to study these sources. We also found that this new phenomenon in AMPs is related to variations in the observed pulse shapes, which in our interpretation shows that the accretion disc–magnetosphere interaction region in AMPs is highly dynamic. In future studies, this finding will help us to utilize AMP pulsations to constrain NS masses and radii more accurately than before. We have also obtained evidence that the majority of ULXs are not powered by accretion onto intermediate-mass BHs, which had been suggested by numerous previous studies. Our analysis instead supports the view that the extreme luminosities of a majority of ULXs are a result of very high mass accretion rate onto “typical” BHs having no more than a few tens of solar masses.