The book presents very recent advances in understanding dark matter, dark energy effects, orbital dynamics of exoplanets around binary stars in non-planar configurations, the orbital dynamics of a moon in the star-planet systems, the orbital dynamics of interstellar interlopers around binary stars, the mechanisms of stellar flares, parameters of white dwarfs atmospheres. For achieving this purpose, the presentation covers a brief history of each topic (including a large number of references), provides detailed derivations of the recent advanced results, and illuminates the physics behind these advanced results. In particular, the book provides explanations of the latest perplexing astrophysical observations of the anomalous absorption of the 21 cm spectral line from the early Universe, as well as of the recent puzzling observational results by the Dark Energy Survey team, who found that the distribution of dark matter is smoother than predicted by Einstein's general relativity. The book presents an overview of dark matter theories from the viewpoint of the Occam razor principle. The book presents also an alternative explanation for the era of the decelerating expansion of the Universe and the era of the accelerated expansion of the Universe - without resorting to dark energy or without introducing new gravitational degrees of freedom. The book presents analytical results on the trajectory of an interstellar interloper in the vicinity of a binary star - the results enabling to determine parameters of the interloper from observations. The fundamental importance of these results is due to the fact that our solar system has been recently visited by two interstellar interlopers: Oumuamua and 2I/Borisov. Further, the book presents advances in understanding mechanisms of stellar flares. By the analysis of the observations, it is shown that flare stars are the most favorable object for studying nonthermal physical phenomena in the visible range. Next, the book presents new results related to white dwarfs. White dwarfs are characterized by very strong magnetic fields: from 100 to 100000 Tesla. In these very strong magnetic fields, electrons in the atmospheric plasmas of white dwarfs move along helical trajectories. It is shown that the neglect of this effect leads to the overestimation of the electron density, deduced from the observed widths of Balmer lines, by several times. In view of these results, it is clear that there is required a very significant revision of the electron density deduced from all observations of hydrogen lines in white dwarfs. Finally, the book presents various types of the astrophysical evidence pointing to the existence of a parallel universe. Totally different kinds of astrophysical observations demonstrated that in our universe there exists a preferred direction. It is shown that there could be a parallel universe in this direction, both the parallel universe and our universe rotating about their barycenter.