The rising resistance and the severe side effects associated with classical anticancer drugs such as cisplatin or mercaptopurines has triggered the development of novel antitumourous agents based on transition metal (e.g. ruthenium, osmium, and gold) organometallic complexes. In parallel, the tools of computational biochemistry have become more and more powerful and can nowadays provide important insights into the mechanistic origins of many systems of medicinal relevance creating a synergetic working environment between experimental and theoretical approaches. In this dissertation, a range of computational methods to study molecules from a quantum mechanical and/or classical mechanical point of view have been applied to characterise key properties of biological drug-target complexes. All of the computational research presented herein has been performed in tight collaboration with experimental groups, which determined X-ray structures of drug-target adducts and characterised the anticancer properties of the synthesised novel anticancer drugs studied here. In particular, the interactions of a small ruthenium-based complex acting as an DNA semi-intercalator,[(eta^6-5,8,9,10-tetrahydroanthracene)Ru(II)(ethylenediamine)X] with X representing a leaving group such as Cl-, were characterised exposing different binding modes of the compounds with free vs. nucleosomal DNA. Additionally, classical molecular dynamics simulations were used to complete a characterisation of the localisation of the linker of a novel binuclear antitumourous compound, that was designed in such a way as to take advantage of newly discovered allosteric pathways present in the nucleosome core particle with the aim of creating a synergistic effect in the uptake the two moieties. In addition to the localisation of the linker, the effects of the bound compound on the allostery of the nucleosome core particle has been explored to compare the mode of action of the newly synthesised binuclear compound with that of non-linked analogue with synergistic properties. Moreover, a similar study has been performed on a herpes virus peptide, (Met-Ala-Pro-Pro-Gly-Met-Arg-Leu-Arg-Ser-Gly-Arg-Ser-Thr-Gly-Ala-Pro-Leu-Thr-Arg- Gly-Ser), binding to the nucleosome core particle to observe whether or not it would be able to trigger allosteric pathways. Finally, some fundamental issues related to the selectivity of transition metal based anticancer compounds for specific binding sites on the nucleosome core particle were tackled by simulating the approach of three different ruthenium and osmium-based complexes, [(eta^6-cymene)Ru(II)(1,3,5-triaza-7-phosphoadamantane)X_2], [(eta^6-cymene)Ru(II)(N-phenylpyridine-2-carbothioamide)X] and [(eta^6-cymene)Os(II)( N-phenylpyridine-2-carbothioamide)X], from the bulk solvent to their bound state with the histone proteins with the help of classical and mixed quantum mechanical/molecular mechanical molecular dynamics simulations. A determination and analysis of the corresponding free energy profiles were used to provide some rationalisation of the observed preference of different compounds for specific binding sites. All the computational studies performed here and the insights gained can provide some guidance for the rational design of anticancer compounds with specific binding properties.