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In this thesis, we investigate the mechanisms driving the self-assembly of peptides and proteins using computational and theoretical tools, always validating our results with experimental measures when possible. In the first part, Chapters 2-5, we focus on the Aβ system, a peptide whose aggregation is intimately linked with the development of Alzheimer's Disease. We begin by simulating the major alloforms of the peptide, Aβ_40 and Aβ_42, demonstrating that the two populate similar disordered ensembles and matching experimental data. Next we investigate how disordered Aβ_42 monomers interact with each other, finding that oligomerisation into amorphous aggregates is driven largely by hydrophobic, non-specific forces. We then move on to probing the aggregation of Aβ_42 into amyloid structures using a native-centric coarse-grained model, and explain the results with a novel Markov state analysis from which we are able to extract structural, kinetic and thermodynamic information on elongation reactions. Finally, we probe the interactions of Aβ_42 monomers with Aβ_42 fibrillar surfaces using a specially designed enhanced sampling scheme, which allows us to obtain enthalpy-driven binding thermodynamics consistent with experiments and to propose major polar binding modes. In the second part of the thesis, Chapters 6 and 7, we model the aggregation of two other self-assembling systems, viruses and a truncated form of the molecular chaperone Hsp70. We first develop a data analysis platform to extract information on the microscopic mechanisms of viral capsid self-assembly from experimental data, synthesising the results from several different systems to draw general evolutionary conclusions about the assembly mechanism. Finally, we model the oligomerisation of Hsp70 thermodynamically and kinetically, showing that its self-assembly is a highly cooperative reaction that is under strong structural constraints.