Increasing volatilities within power transmission and distribution force power grid operators to amplify their use of communication infrastructure to monitor and control their grid. The resulting increase in communication creates a larger attack surface for malicious actors. Indeed, cyber attacks on power grids have already succeeded in causing temporary, large-scale blackouts in the recent past. In this paper, we analyze the communication infrastructure of power grids to derive resulting fundamental challenges of power grids with respect to cybersecurity. Based on these challenges, we identify a broad set of resulting attack vectors and attack scenarios that threaten the security of power grids. To address these challenges, we propose to rely on a defense-in-depth strategy, which encompasses measures for (i) device and application security, (ii) network security, (iii) physical security, as well as (iv) policies, procedures, and awareness. For each of these categories, we distill and discuss a comprehensive set of state-of-the art approaches, and identify further opportunities to strengthen cybersecurity in interconnected power grids.