Graphene is a single layer of carbon atoms arranged in honeycomb lattice. Over a long period of time it was treated as a hypothetical material to understand the properties of other allotropes of carbon, such as graphite, carbon nanotube etc. Half decade back, a single layer of graphene was finally isolated and since then the field has observed a flurry of activities. Low energy excitations in graphene are massless Dirac Fermions and quantum electrodynamic effects can be observed at room temperature in graphene, which makes it very popular among the condensed matter community. In addition graphene also shows many interesting mesoscopic effects, which is the focus of the present work. We study the electronic, magnetic and structural properties of the graphene nanostructures. The entire thesis based on the results and findings obtained from the present investigation is organized as follows. Chapter 1: provides a general introduction to the properties of graphene and graphene based nanostructures. Chapter2:describes the theoretical tools used in this thesis to investigate the properties of graphene nanoribbons. The first two chapters are meant to give the reader an overview about the field of graphene and a few of the computational techniques commonly used to investigate the properties of graphene. The following chapters are the new findings reported in this thesis. Chapter3:shows how zigzag graphene nanoribbons respond in a non-linear fashion when edges are subjected to some external potential such as magnetic field. Such response originates from the edge states present in zigzag ribbons and thus not observed in armchair nanoribbons. In the limit of ribbon width W→∞, an edge magnetic field produces a moment of ~ 1/3 per edge atom even for an infinitesimally small field, which is clearly a signature of non-linear response. Response of a finite width nanoribbon is size dependent and also depends on ln(V), the applied field. This is akin to Weber-Fechner law of audio visual perceptions. It is interesting to note that nature does provide a “quantum realization” of this in the form of biological sensing organs like the ear and eye. The magnetic response is found to scale inversely with the ribbon width. Chapter4:deals with the magnetic properties of the zigzag graphene nanoribbon. This is also a special property of the geometry of the zigzag edges and not observed in armchair nanoribbons. Our investigation reveals that the electron-electron repulsion (Hubbard U) energy creates a delta function like edge magnetic field in zigzag graphene nanoribbons. Starting from this, magnetic properties of zigzag graphene nanoribbons can be qualitatively and quantitatively explained from the non-linear response of zigzag nanoribbons. Zigzag graphene nanoribbons can exist in two possible ‘magnetic states’: antiferro (AF) where the two opposite edges have antiparallel magnetic moment and ferro (FM) where moment is parallel in the two opposite edges. First we describe the properties of undoped zigzag nanoribbons. They have AF ground state. Continuum theory can explain the size dependent bandgap and magnetic moment of the ground state. We present the first explicit derivation of the gap. Then we show that hole doping can change the ground state to FM, which is metallic. Thus the system has the property of magnetoresistance, which can be exploited by doping zigzag graphene nanoribbons externally with some gate voltage or internally by some electron acceptor element, such as boron. The critical doping for transition depends inversely with the ribbon width. We have found that the ‘phase transition’ on hole doping is a common phenomena for zigzag terminated nanostructures, such as hexagonal nanodots. Chapter5:discusses the effects of random edge shapes and random potential (Anderson disorder) on the magnetic properties of zigzag graphene nanostructures. Defects and disorders in the form of edge shape randomness and random potentials arising from substrate are very common in graphene. Our study reveals that edge state magnetism is very robust to shape randomness of the terminating edges of nanostructures; as long as there are three to four repeat units of a zigzag edge, the edge state magnetism is preserved. We also discover some “high energy” edges (ones where the edge atoms have only one nearest neighbor) can have very large moments compared to even the zigzag edges. Edge magnetism is also found to be robust to relatively small Anderson disorders, because a slowly varying small potential does not scatter the edge states. Chapter6:reveals how edge functionalization by O atom and OHgroup changes the properties of the zigzag graphene nanoribbons. Functionalization by various different molecules is a very popular method of tuning the properties of graphene. We have shown that it is possible to tune the properties of zigzag graphene nanoribbons by edge functionalization. Further, we have found that structures with clustered functionalization leads to “spatially” varying electronic structure, which can lead to interesting possibilities for electronic devices. Chapter7:describes structural stability, electronic and magnetic properties of graphene nanoribbons in presence of topological defects such as Stone-Wales defects. Our study reveals that the sign of stress induced by a SW defect in a graphene nanoribbon depends on the orientation of the SW defect with respect to the ribbon edge and the relaxation of the structure to relieve this stress determines its stability. Local warping or wrinkles arise in graphene nanoribbon when the stress is compressive, while the structure remains planar otherwise. The specific consequences to armchair and zigzag graphene nanoribbon can be understood from the anisotropy of the stress induced by a SW defect embedded in bulk graphene. We also have found localized electronic states near the SW defect sites in a nanoribbon. However, warping results in delocalization of electrons in the defect states. We have observed that, in zigzag graphene nanoribbons magnetic ordering weakens due to the presence of SW defects at the edges and the ground state is driven towards that of a nonmagnetic metal.