The development and approval of new drugs is an expensive process. The total cost for the approval of a new compound is on average 1.0 - 1.2 billion dollars and the entire process lasts about 12 - 15 years. The main difficulties are related to poor pharmacokinetics, lack of efficacy and unwanted side effects. These problems have naturally led to the question if new and alternative methodologies can be developed to find reliable and low cost alternatives to existing practices. Nowadays, computer-assisted tools are used to support the decision process along the early stages of the drug discovery path leading from the identification of a suitable biomolecular target to the design/optimization of drug-like molecules. This process includes assessments about target druggability, screening of molecular libraries and the optimization of lead compounds where new drug-like molecules able to bind with sufficiently affinity and specificity to a disease-involved protein are designed. Existing computational methods used by the pharmaceutical industry are usually focused on the screening of library compounds such as docking, chemoinformatics and other ligand-based methods to predict and improve binding affinities, but their reliable application requires improvements in accuracy. New quantitative methods based on molecular simulations of drug binding to a protein could greatly improve prospects for the reliable in-silico design of new potent drug candidates. A common parameter used by medicinal chemists to quantify the affinity between candidate ligands and a target protein is represented by the free energy of binding. However, despite the increased amount of structural information, predicting binding free energy is still a challenge and this technique has found limited use beyond academia. A major reason for limited adoption in the industry is that reliable computer models of drug binding to a protein must reproduce the change in molecular conformations of the drug and protein upon complex formation and this includes the correct modelling of weak non-covalent interactions such as hydrogen bonds, burials of hydrophobic surface areas, Van der Waals interactions, fixations of molecular degrees of freedom solvation/desolvation of polar groups and different entropy contributions related to the solvent and protein interactions. For several classes of proteins these phenomena are not easy to model and often require extremely computationally intensive simulations. The main goal of the thesis was to explore efficient ways of computing binding affinities by using molecular simulations. With this aim, novel software to compute relative binding free energies has been developed. The implementation is based on alchemical transformations and it extended a preexisted piece of software Sire, a molecular modeling framework, by using the OpenMM APIs to run fast molecular dynamics simulations on the latest GPGPU technology. This new piece of software has equipped the scientific community with a flexible and fast tool, not only to predict relative binding affinities, but also a starting point to develop new sampling methods for instance hybrid molecular dynamics and Monte Carlo. The implementation has been validated on the prediction of relative hydration free energy of small molecules, showing good agreement with experimental data. In addition, non-additive effects to binding affinities in series of congeneric Thrombin inhibitors were investigated. Although excellent agreement between predicted and experimental relative binding affinities was achieved, it was not possible to accurately predict the non-additivity levels in most of the examined inhibitors, thus suggesting that improved force fields are required to further advance the state-of-the art of the field.