The transition to a climate neutral energy system relies on an increasing share of renewable energy sources in European electricity grids. As the production of renewable energy sources is inherently variable, flexibility requirements to balance supply and demand are expected to grow in the years to come. In this work, we study the flexibility needs in the 2030 and 2050 European power system using the METIS energy system model. We find flexibility requirements to increase significantly at different timescales, with variable renewable energy output as the main driver. We identify those technologies, including storage, which may offer flexibility solutions and we assess the economic value of financial arbitrage for such flexibility technologies in the spot market. We further study, in relation to storage investment costs and available interconnection capacity, the optimal combination of electricity storage solutions to accommodate flexibility needs in future European power systems.