Ascribing to the recent trends of market-driven electricity production and increased deployment of non-dispatchable renewables globally, several researchers have suggested the use of variable speed technology to improve the operational flexibility and efficiency of conventional Francis turbines. While this technology has proven extremely useful for contemporary operation of Reversible Pump Turbines (RPT), still, there is no example in existence of its application to a conventional Francis turbine. So far, one of the main reasons for this appears to be the small efficiency gain that is not enough to overcome the losses from the devices that enable speed variation. However, with the recent developments in full-size frequency converters, as well as the plans for operation of Francis turbines in a much wider operating range in the future, this is expected to change. In this thesis, the main objective is twofold. The first part was to provide a more detailed analysis of the efficiency gains and pressure fluctuations aspects that the technology could provide for low specific speed machines. In the second part, methods for numerical optimization are used to conduct a detailed parametric study on the possibility to improve the variable speed performance of a reference turbine. The main accent is placed on the point that a turbine, which is meant to be operated at variable speeds exclusively, should be designed and optimized for that purpose from day one, and this may not necessarily be equal to the design philosophy of a synchronous speed representative. The Waterpower laboratory at the Norwegian University of Science at Technology in Trondheim, Norway, has provided a unique opportunity for the early experimental work that the author has conducted for the thesis. Efficiency and pressure pulsation measurements were done for two runners of a comparable specific speed, namely one RPT design and one splitter-bladed Francis design, that could be installed in the same turbine for a direct comparison of the variable speed performance. This study provided an essential basis for the further work, suggesting that the level of efficiency gain from the variable speed operation is greatly dependent on the hydraulic design of the runner. Additionally, it is shown that when operating at rotational speeds specifically optimized for maximum efficiency, the amplitudes of the pressure pulsations in both runners were either reduced or stayed at the same level as for the synchronous speed operation. To provide an insight on the influence that the runner design has on the shape of the hill chart, as well as to make an educated decision on which design parameters to be used for the optimization process later, a theoretical model was developed and studied. In this step, even though only minimal geometric information for the turbine was used as the input, the simple one-dimensional model was able to predict the general characteristics of both runners that were experimentally investigated. It is demonstrated that the geometry of the runner at the inlet and the outlet (i.e. the width of the meridional channel, the metal angles of the blade and the ratio between the inlet and outlet diameters) have the most dominant effects on the performance at off-design operating conditions. A parametric environment for designing of turbine runners has been developed by the author and used for optimization of a replacement runner for the Francis turbine model installed in the Waterpower laboratory. Relying exclusively on the use of Bézier curves, the constrained design space of the runner is described with 15 free parameters that provide a wide geometric variation at the critical zones pointed out by the one dimensional study. Together with this, a suitable objective function was also developed and defined to secure a trustworthy steering towards an improved variable speed performance of the replacement runner. Results from calculations with Computational Fluid Dynamics were used to train surrogate models of the turbine performance and were used to explore the sensitivity of the design parameters and selection of a tradeoff design that fulfills the optimization criteria. Surprisingly, it was found that the shape of the hill chart cannot be altered significantly and that most of the parameters that were considered had their main effects on the level of the peak efficiency and its position in the hill chart area. The optimal design was selected as a tradeoff between the variable speed objectives, that also happened to outperform the reference at synchronous speed operation as well. Future research regarding the hydraulic design of variable speed Francis turbines should focus less on the detailed geometry of the blade and more on the global sizing of the turbine, which is done by varying the rotational speed, the ratio between the inlet and outlet diameters of the runner and the inlet width. In that case, however, more aspects will have to be checked, such as the overall size and price of the turbine and the cavitation performance.