To overcome the worldwide challenges of climate change, photovoltaics is foreseen to play a significant role in the world electricity production. Nowadays, single junction crystalline silicon (c-Si) based solar cells hold the largest share of the global photovoltaic market. To achieve their maximum energy conversion efficiency, carrier-selective passivating contacts have been demonstrated to be the most promising technologies. Among them, silicon heterojunction (SHJ) solar cells are expected to further reduce the cost of photovoltaics. Indeed, they have achieved record efficiency thanks to high c-Si bulk quality and excellent full area surface passivation. However, despite remarkable performance, the material layers used to build SHJ cells are at the origin of significant trade-offs between electrical and optical properties. In particular, the properties coupling arising between the layers can cause significant electrical transport losses, strongly capping the SHJ device performance. The aim of this thesis is to accurately describe, quantify, and mitigate the electrical losses occurring in SHJ solar cells by means of advanced methodologies and characterization methods. Key outcomes of this work are fourfold. First, we introduce a generalized and unambiguous description of the concept of contact using the terminology of shell. Then, we propose a characterization methodology using two new approaches named top-down and bottom-up, designed to independently track the properties of the material layers used to build the shells, as well as the performance of the solar cells incorporating the shells. This was shown to allow one to accurately investigate and mitigate the electrical and optical losses affecting solar cells. Secondly, we present transfer length method (TLM) measurements under variable illumination as a novel advanced characterization method to study the electrical transport losses in SHJ devices. We demonstrated (i) that illumination, i.e., the injected carrier density inside the c-Si bulk, has a strong impact on the contact resistivity (rc), (ii) the importance of measuring rc under maximum power point conditions for a relevant characterization of transport losses, and (iii) how the dependence of the rc on the injected carrier density enables the comparison of the illumination response of different SHJ shells. In addition, we conducted preliminary investigations of the applicability of TLM under illumination to measure p-type shell parts on n-type c-Si wafers. Thirdly, we engineered various material layers and studied their properties. Multilayering the thin hydrogenated silicon and the TCO layers demonstrated a high potential for electrical losses mitigation by (i) improving the properties coupling in the shell, (ii) addressing the constraints of the device architecture, and (iii) mitigating the trade-offs impacting solar cell performance. Finally, the last outcome is the integration of the most promising shells in actual baseline solar cells. This resulted in fill-factors up to 83.2%, for 6-inch SHJ cells, and in efficiencies up to 25.45% for SHJ IBC cells. Finally, combining optimal electrical and optical shell properties, an impressive efficiency of 24.24% was achieved using an aluminium-doped zinc oxide as rear TCO. The advanced methodologies and characterization methods presented in this work are foreseen to drive the understanding and optimization of other solar cell technologies such as, e.g. TOPCon or tandem solar cells.