Lipid molecules are amphiphilic in nature consisting of a hydrophilic head group and hydrophobic hydrocarbon tails. The lipid bilayer consists of two layers of lipid molecules arranged with their hydrophobic tails facing each other and their hydrophilic head groups solvated by water. Lipid bilayers with hydrophilic polymer chains grafted onto the head groups have applications in various fields, such as stabilization of liposomes designed for targeted drug delivery, synthesis of supported bilayers for biomaterial applications, surface modification of implanted medical devices to prevent biological fouling and design of in vitro biosensors. The focus of this thesis lies in understanding the effects of polymer grafting on the thermodynamics and mechanical properties of lipid bilayers. Dissipative particle dynamics (DPD) has evolved as a promising method to study complex soft matter systems. The basic DPD algorithm, and its implementation are discussed in Chapter 2 of this thesis. It is important to achieve a tensionless state while studying phase transitions and deducing the mechanical properties of the bilayer. We proposed a modification of the Andersen barostat which can be incorporated in a DPD simulation to achieve the tensionless state as well as carry out simulations at a prescribed tension. In Chapter 3 of this thesis the effect of polymer grafting on single tailed lipid bilayers is studied. Simulations are carried out by varying the grafting fraction, Gf, defined as the ratio of the number of polymer molecules to the number of lipid molecules. At lowGf, the bilayer shows a sharp transition from the gel (Lβ) to the liquid crystalline (Lα) phase. This main melting transition temperature is lowered as Gf is increased. Corresponding to this, an increase in the area per head group is also observed. Above a critical value of Gf the interdigitated, LβI phase is observed prior to the main transition for the longer lipid tails. The analysis for two tailed lipids as a function of polymer chain length is extensively studied in Chapter 5. For the case of two tailed lipids, an intermediate interdigitated phase was not observed and the decrease in the melting temperature is more pronounced as the length of the polymer chain is increased. The scaling for fractional change in the area per head group, as well as the decrease in transition temperature as a function of polymer grafting are in good agreement with mean field theory predictions. The bending modulus (k) and area stretch modulus (kA) are essential for determining the shape and the mechanical stability of biological cells or lipid based vesicles. In simulations, the bending modulus k is evaluated from the Fourier transform of the out-of-plane fluctuations of the bilayer mid-plane. In Chapter 4 of this thesis, we illustrate that a surface representation based on Delanuay triangulation provides a robust parameter free representation of the bilayer surface. By evaluating the bending modulus for single tail lipids of different tail lengths, the continuum scaling relation d2 is verified. To our knowledge this is the first systematic investigation and verification of this scaling relationship using computer simulations. Using the continuum relation, =kAd2/ we find that α depends weakly on the tail lengths of the bilayer. Nevertheless we illustrate that a value of α=130 can be used to reliably estimate the bending modulus from the area stretch modulus for polymer free bilayers. Using our method, we are also able to capture the low q scalings and obtain the bending modulus of the gel (Lβ) phase. Grafted polymer was found to increase the value of the bending modulus for single tail lipids. Although the presence of polymer directly increases the area per head group, the suppressed height fluctuations dominate and the bending modulus increases for the single tail lipids. For two tail lipids a small decrease in the bending modulus was observed at low grafting fractions and short polymer chains. For large polymer lengths the bending modulus was found to increase monotonically.