The average dissipation generated during a slow thermodynamic process can be characterised by introducing a metric on the space of Gibbs states, in such a way that minimally-dissipating protocols correspond to geodesic trajectories. Furthermore, the average dissipation is proportional to the work fluctuations for classical systems (which follows from the fluctuation-dissipation relation (FDR)), so that minimising dissipation also minimises fluctuations. In this talk, I will explain how this geometric picture is modified in the quantum regime. First, I will show that slowly driven quantum systems violate the classical FDR whenever quantum coherence is generated along the protocol, implying that quantum non-commutativity prohibits finding slow protocols that minimise both dissipation and fluctuations simultaneously. Instead, we develop a quantum geometric framework to find processes with an optimal trade-off between the two quantities. Then, I will apply these geometric techniques to optimise the power output of a finite-time Carnot engine, and prove that the maximal power becomes proportional to the heat capacity of the working substance. Since the heat capacity can scale supra-extensively with the number of constituents of the engine (e.g. in a phase transition point), this enables us to design many-body heat engines reaching Carnot efficiency at finite power per constituent in the thermodynamic limit. This talk is based on: arXiv:1810.05583, arXiv:1905.07328, and arXiv:1907.02939.