The topic of this thesis is the study of the vertical structure of the disk galaxies and their dark matter halos through theoretical modeling and numerical calculations. The basic theoretical model of the galactic disk used involves gravitationally-coupled stars and gas under the force-field of a dark matter halo; the disk is rotationally-supported in the plane and pressure-supported perpendicular to the plane of the galaxy. The first part of the thesis involves evaluating the vertical structure of stars and gas in normal as well as dwarf spiral galaxies. The second part of the thesis deals with probing the dark matter halo density profiles of disk galaxies using both the observed rotation curve and the H i scale height data. Following is the layout of the thesis. Chapter 1 gives a general introduction to the topic of vertical structure of spiral galaxies and their dark matter halos, followed by a broad overview of the theoretical development of the topic and ends with highlighting the motivation and challenges met in this thesis. Chapters 2 & 3 deal with the vertical structure of stars and gas in galaxies, Chapters 4-6 focus on obtaining the dark matter halo density profiles of disk galaxies from the observed rotation curve and the H i scale height data whereas Chapter 7 is devoted to the summary of results and future research plans. Vertical structure of stars and gas in galaxies The vertical thickness of the stars and the gas, namely atomic hydrogen (H i) and molecular hydrogen (H2) in a spiral galaxy, is crucial in regulating the disk dynamics close to the mid-plane, especially in the inner galaxy. However, measuring it observationally is not in general practicable due to the limitations of astronomical observations, and often impossible as in the case of face-on galaxies. Therefore, it is imperative to develop a theoretical model of the galaxy which can predict the thickness of the disk components by using as input parameters the physical quantities, which are more observationally-amenable compared to the disk thickness. The vertical thickness of the disk components is determined by a trade-off between the upward kinetic pressure and the net downward gravitational pull of the galaxy. The fraction of the disk mass due to the stars is an order of magnitude higher than that of the gas in ordinary spiral galaxies, and therefore the gas contribution to the disk gravity is ignored in general. We have developed a multi-component model of gravitationally-coupled stars, HI and H2 subjected to the force-field of an external dark matter halo, and conclusively demonstrated the importance of the inclusion of gas gravity in explaining the steep vertical stellar distribution observed in galaxies. These apart, this model does not implicitly assume a flat rotation curve for the galaxy and therefore is applicable in general to obtain the thickness of stars and gas in dwarfs (with linearly rising rotation curves) as well as in ordinary spirals. In Chapter 2, we investigate the origin of the steep vertical stellar distribution in the Galactic disk. One of the direct fall outs of our above model of the galaxy, which incor¬porates the self-gravity of the gas unlike the earlier theoretical models, lies in explaining the long-standing puzzle of the steep vertical stellar density distribution of the disk galax¬ies near the mid-plane. Over the past two decades, observations revealed that the vertical density distribution of stars in galaxies near the mid-plane is substantially steeper than the sech2 function that is expected for a self-gravitating system of stars under isothermal ap¬proximation. However, the physical origin for this has not been explained so far. We have clearly demonstrated that the inclusion of the self-gravity of the gas in the dynamical model of the Galaxy solves the problem even under the purview of isothermal approximation for the disk components. Being a low dispersion component, the gas resides closer to the mid¬plane compared to the stars, and forms a thin, compact layer near the mid-plane, thereby strongly governing the local disk dynamics. This novel idea, highlighting the significance of gas gravity has produced substantial impact on the field and triggered research activities by other groups in related areas of disk dynamics. The strong effect of the gas gravity on the vertical density profile of the stellar disk indicates that it should also bear its imprint on the Milky way thick disk, as the epoch of its formation 109 years ago is marked by a value of gas fraction, almost an order of magnitude higher than its present day value. Interest-ingly, the findings of the upcoming Gaia mission can be harnessed to verify this theoretical prediction. It may also hold the clue as to the reason behind the absence of thick disk in superthin galaxies. In Chapter 3, we use the same model to theoretically determine the H i vertical scale heights in the dwarf galaxies: DDO 154, Ho II, IC 2574 & NGC 2366 for which most of the necessary input parameters are available from observations. We stress the fact that the observational determination of the gas thickness in these dwarf irregulars is not viable. Nevertheless, it is important to estimate it theoretically as it plays a crucial role in calculating the star-formation activities and other related phenomena. However, two vital aspects have to be taken care of while modeling these dwarf galaxies. Firstly, the mass fraction in gas in these galaxies is comparable to that of the stars, and hence the gas gravity cannot be ignored on any account unlike in the case of large spirals. Secondly, dwarf galaxies have a rising rotation curve over most of the disk unlike the flat rotation curves of ordinary spirals. Both these factors have been considered in developing our model of the dwarf galaxies. We find that three out of the four galaxies studied show a flaring of their H i disks with increasing radius, by a factor of a few within several disk scale lengths. The fourth galaxy (Ho II) has a thick H1 disk throughout. A comparison of the size distribution of H1 holes in the four sample galaxies reveals that of the 20 type 3 holes, all have radii that are in agreement with them being still fully contained within the gas layer. Probing the dark matter halo profiles of disk galaxies The next part of the thesis involves the dynamical study of the shapes and density profiles of galactic dark matter halos using observational constraints on our theoretical model of a spiral galaxy. The density distribution of the dark matter halo is generally modeled using the observed rotation curve of the spiral galaxies. The rotational velocity at any radius is determined by the radial component of the net gravitational force of the galaxy, which, however, is weakly dependent on the shape of the dark matter halo. Therefore, one cannot trace the dark matter halo shape by the observed rotation curve alone. The vertical thickness of the stars and gas, on the other hand, is strongly dependent on the flattening of the dark matter halo, and therefore the observed gas thickness can be used as a diagnostic to probe the halo shape. In this thesis, we have used the double constraints of the rotation curve and the H i thickness data to obtain the best-fit values of the core density, core radius and the vertical-to-planar axis ratio (or flattening) of the dark matter halos of our largest nearby galaxy Andromeda (or M31), a low-surface brightness (LSB) superthin galaxy UGC 7321 and to study the dark matter halo shape of our Galaxy. In Chapter 4, we study the dark matter halo of M31 or Andromeda, the largest nearby galaxy to the Milky Way. We find that M31 has a highly flattened isothermal dark matter halo with the vertical-to-horizontal axis ratio equal to 0.4, which interestingly lies at the most oblate end of the halo shapes found in cosmological simulations. This indicates that either M31 is a unusual galaxy, or the simulations need to include additional physics, such as the effect of the baryons, that can affect the shape of the halo. This is quite a remarkable result as it challenges the popular practice of assuming a spherical dark matter halo in the dynamical modeling of the galaxy In Chapter 5, we have applied this technique to the superthin galaxy UGC 7321. Su¬perthins are somewhat the “extreme” objects in the local Universe because of their high gas fraction and absence of a thick disk component. It is interesting to analyze their so-called extreme characteristics in the light of the physical mechanisms which determined them to understand better the properties of ordinary spirals. We find that UGC 7321 has a spher¬ical isothermal halo, with a core radius almost equal to the disk scale length. This reveals that the dark matter dominates the dynamics of this galaxy at all radii, including the inner parts of the galaxy. This is unlike the case for the large spiral galaxies, where the core radius is typically about 3-4 disk scale lengths. Interestingly, the best-fit halo core density and the core radius are consistent, with deviations of a few percent, with the dark matter fundamental plane correlations, which depict the systematic properties of the dark matter halo in late-type and dwarf spheroidal galaxies. This apart, a high value of the gas velocity dispersion is required to get a better fit to the H i scale height data, although the superthin nature of the stellar disk implies a dynamically cold dynamic galactic disk. However, it explains the low star-formation rates in these galaxies since the Toomre Q criterion (Q < 1) for instability is less likely to be satisfied, and hence the disk is liable to be more stable to star formation. In Chapter 6, we investigate the shape of the dark matter halo in the outer Galaxy. We find that the halo is prolate, with the vertical-to-planar axis ratio monotonically increasing to 2.0 at 24 kpc, or 8 radial disk scale lengths. The resulting prolate-shaped halo can explain several long-standing puzzles in galactic dynamics, for example, it permits long-lived warps thus explaining their ubiquitous nature. It also imposes novel constraints on the galaxy formation models. Finally, in Chapter 7, the thesis is concluded with a summary of the main results and a brief discussion of the scope for future work.