Graphene – a two-dimensional honeycomb lattice of sp2-bonded carbon atoms – has been attracting a great deal of research interest since its first experimental realization in 2004, due to its various novel properties and its potential for applications in futuristic nanodevices. Being the fundamental building block for carbon allotropes of other dimensionality, it can be stacked to form 3d graphite or rolled into 1d nanotube. Graphene is the thinnest known material in the universe, and one of the strongest materials ever measured in terms of its in-plane Young modulus and elastic stiffness. The charge carriers in graphene exhibit giant mobility as high as 20 m2/Vs, have almost zero effective mass, and can travel for micrometers without scattering even at ambient conditions. Graphene can sustain current densities six orders of magnitude higher than that of copper, shows record thermal conductivity and stiffness, is impermeable to gases, and renders easy accessibility to optical probes. Electron transport in graphene is described by a Dirac-type equation, which allows the investigation of “relativistic” quantum phenomena in a benchtop experiment. This results in the observation of a number of very peculiar electronic properties from an anomalous quantum Hall effect to Kien paradox and the absence of localization. All these enticing features make this material an excellent candidate for application in various electronic, photonic and optoelectronic devices. For instance, its ballistic ambipolar transport and high carrier mobility are the most useful traits for making ultrafast and low-power electronic devices. Its high surface area shouldmake it handy in manufacturing tough composite materials. The extreme thinness of graphene could also lead to more efficient field emitters that release electrons in the presence of strong electric fields. Its robustness and light weight are useful for micromechnical resonators. The tunability of its properties could make it possible to build so-called spin-valve transistors, as well as ultra-sensitive chemical detectors. Many of such applications of graphene require tuning of its properties, which can be achieved by varying the number of layers or/and by doping. There are several ways to dope graphene: (i)electrochemically gated doping, (ii)molecular charge-transfer doping, and (iii) substitutional doping by atoms like Boron or Nitrogen.Moreover, for graphene, a zero band gap semiconductor in its pristine form, to become a versatile electronic device material it is mandatory to find means to open up a band gap and tune the size of the band gap. Several strategies have been adopted to engineer such a band gap in graphene in a controlled way. Some of these are based on the ability to control the geometry of graphene layers, some use graphene-substrate interactions, while others are based on chemical reactions of atoms or molecules with the graphene layer. Motivated by these considerations, in this thesis we present a systematic and thorough study of the structural, electronic and vibrational properties of graphene and their dependence on the number of layers, and on doping achieved electrochemically, molecularly and substitutionally, using first principles density functional theory (DFT). In Chapter 1, we give an introduction to the hitherto beguiling world of graphene. Here, we briefly discuss the structure, novel properties and potential applications of graphene, and the motivation for this thesis. In Chapter 2, an overview of the DFT formalism adopted here is given. We clearly state the theorems of the formalism and the approximations used when performing calculations. We succinctly explain how the various quantities like total energies, forces, stresses etcetera are calculated within this formalism. We also discuss how phonon frequencies, eigenvectors, electron-phonon couplings are obtained by using density functional perturbation theory (DFPT), which calculates the full dynamical matrices through the linear response of electrons to static perturbations induced by ionic displacements. Calculations are done first using a fully ab-initio approach within the standard Born-Oppenheimer approximation, and then time-dependent perturbation theory is used to explore the effects of dynamic response. In Chapter 3, using such first-principles density-functional theory calculations, we determine the vibrational properties of ultra-thin n(1,2,...,7)-layer graphene films and present a detailed analysis of their zone-center phonons. We present the results (including structural relaxations, phonons, mode symmetries, optical activities) for bulk Graphite, single-layer graphene and ultrathin n-layer graphene films. and discuss the underlying physics of our main results together with a pictorial representation of the phonon modes. We demonstrate that a low-frequency (∼ 112 cm−1 ) optical phonon with out-of-plane displacements exhibits a particularly large sensitivity to the number of layers, although no discernible change in the interlayer spacing is found as n varies. Frequency shifts of the optical phonons in bilayer graphene are also calculated as a function of its interlayer separation and interpreted in terms of the inter-planar interaction. The surface vibrational properties of n-layer graphene films are presented in Chapter 4, which renders a detailed and thorough analysis of all the surface phonon modes by determining, classifying and identifying them accurately. The response of surface modes to the presence of adsorbed hydrogen molecules is determined. As an illustrative adsorbate, hydrogen is chosen here mainly because of its huge importance in fuel cell technology and as a molecular sensor. We demonstrate that a doubly degenerate surface phonon mode with low-frequency (~ 35cm−1)exhibits a particularly large sensitivity to the adsorption of hydrogen molecules, as compared to other surface modes. Futhermore, we show that a low-frequency (108.8 cm−1)bulk-like phonon with out-of-plane displacements is also very sensitive and gets upshifted by as much as 21 cm−1 due to this adsorption. In Chapter 5, we determine the adiabatic frequency shift of the and phonons in a monolayer graphene as a function of both electron and hole doping. The doping is simulated here to correspond to electrochemically gated graphene. Compared to the results for the E2g -Γ phonon (Raman G band), the results for the phonon are dramatically different, while those for the phonon are not so different. Furthermore, we calculate the frequency shifts, as a function of the charge doping, of the (K + ΔK) phonons responsible for the Raman 2D band –a key finger print of graphene, where [ΔK] is determined by the double resonance Raman process. Doping graphene with electron donating or accepting molecules is an interesting approach to introduce carriers into it, analogous to electrochemical doping accomplished in graphene when used in a field-effect transistor. In Chapter 6, we use first-principles density-functional theory to determine changes in the electronic structure and vibrational properties of graphene that arise from the adsorption of aromatic molecules such as aniline and nitrobenzene. Identifying the roles of various mechanisms of chemical interaction between graphene and the adsorbed molecules, we bring out the contrast between electrochemical and molecular doping of graphene. Our estimates of various contributions to shifts in the Raman active modes of graphene with molecular doping are fundamental to the possible use of Raman spectroscopy in (a)characterization of the nature and concentration of carriers in graphene arising from molecular doping, and (b) graphene-based chemical sensors. Graphene doped electrochemically or through charge-transfer with electron-donor and acceptor molecules, shows marked changes in electronic structure, with characteristic signatures in the Raman spectra. Substitutional doping, universally used in tuning properties of semiconductors, could also be a powerful tool to control the electronic properties of graphene. In Chapter 7, we present the structure and properties of boron and nitrogen doped graphenes, again using first-principles density functional theory. We demonstrate systematic changes in the carrier-concentration and electronic structure of graphenes with B/N-doping, accompanied by a stiffening of the G-band and change of the defect related D-band in the Raman spectra. Such n/p -type graphenes obtained without external fields or chemical agents should find device applications.