Thermonuclear fusion is a potentially clean and limitless energy source that can substantially change the current global electricity generation mix, which is highly dependent on limited fossil fuels. This thesis contributes to the development of fusion energy with experiments on the TCV tokamak, addressing power exhaust in the divertor, which remains a major issue for a fusion tokamak reactor. A large fraction of the heating power needed to keep the plasma at the parameters required for fusion reactions, is continuously transported by the plasma across closed magnetic surfaces into a thin layer with open field lines surrounding the plasma, the Scrape-Off Layer (SOL), where it follows magnetic field lines towards the divertor targets. Unmitigated target peak heat fluxes in a fusion reactor are projected to greatly exceed available material limits. While acceptable target conditions may be possible by operating the divertor in a detached regime, a degradation of core plasma performance may ensue. A unification of a performant core and a detached divertor may be possible using alternative magnetic configurations to the standard Single-Null (SN). This thesis assesses the power exhaust properties over an unprecedented range of alternative magnetic configurations with the goals of both reducing the power exhaust challenge through divertor geometry modifications and improving the current understanding of SOL transport physics. TCV's infrared thermography systems are extensively used to measure divertor target heat fluxes, fitted to extract the SOL heat power width $\lambda_{q,u}$ and the spreading factor $S_u$. Power sharing between divertor targets significantly varies with the plasma and/or divertor geometry, mostly interpreted as the effect of parallel electron heat conduction. The $\lambda_{q,u}$ is found sensitive to properties of the plasma core, shape of the plasma and the divertor, and is consistent with empirical cross-machine scalings. The dependence on plasma current can be mostly understood with a model based on the competition between parallel and perpendicular diffusive transport. Poloidal asymmetries and field direction effects indicate that more physics is needed. The $S_u$ scales with the inverse of the target flux expansion. In the Low-Field-Side Snowflake Minus configuration, the secondary x-point enhances cross-field transport in the divertor, consistently with turbulence simulations. At the L- to H-mode transition, cross-field transport reduces in the main and divertor-SOL, suggesting that the edge transport barrier propagates into the SOL. The upstream $\lambda_{q,u}$ between Type-I ELMs agrees with cross-machine scaling predictions only with the toroidal magnetic field as regression parameter, revealing that this field dependence is vital. A comparison of the ELM power deposition duration in TCV and JET supports the hypothesis of it increasing with the parallel connection length, and the ELM peak parallel energy fluence is consistent with a cross-machine scaling, supporting current extrapolations to ITER. Divertor geometry variations reveal that the ELM power width scales with the inverse of the target flux expansion. These results provide unique input for testing models required to reliably extrapolate to the divertor of a fusion reactor, and may have significant implications in the optimization of its magnetic configuration.