Today, the development of information and communications technologies have changed the utility landscape dramatically. In particular, electricity distribution networks rely heavily on a multitude of intelligent systems and devices that communicate among each other in much more advanced ways than in the past. As the Smart Grid is becoming nowadays a critical component in the electricity delivery system, it is important to make sure the grid is equipped with adequate security mechanisms that are able to guarantee its reliable operation and real-time information exchange within the power infrastructure. Therefore, in this paper we analyze critical cybersecurity aspects associated with smart grid services, including previous cyber-attack cases on smart grids, potential vulnerabilities/threats, and advanced cybersecurity strategies for smart grids with technical and management measures. Ultimately, while the service providers should continuously enhance the traditional security measures such as authentication, access control, authorization, data encryption, public key infrastructure (PKI), firewalls, log analysis, intrusion detection systems, and network security protocols, we propose that the advanced technical measures should 1) make smart grids survivable even under cyber attacks and internal failures; 2) employ a defense-in-depth approach; 3) employ a defense-in-depth approach; and 4) provide more scalable security measures. Furthermore, we also propose that the advanced management measures should 1) establish a cybersecurity governance strategy; 2) develop a strong incident response plan; 3) cultivate a culture of security; 4) employ a public-private partnership approach; and 5) comply with widely recognized security standards.