Modern critical infrastructures such as Smart Grids (SGs) rely heavily on Information and Communication Technology (ICT) systems to monitor and control operations and states within large-scale facilities. The potential offered by SGs includes an effective integration of renewables, a demand-response action and a dynamic pricing system. The increasing use of ICT for the communication infrastructure of modern power systems offers advantages but can give rise to cyber attacks that compromise the security of the SG. To deal efficiently with the security concerns of SGs, a survey of the different attacks that consider the physical as well as the cyber characteristics of modern power grids is required. In the present paper, first the specific differences between SGs with respect to both Information Technology (IT) systems and conventional energy grids are discussed. Thereafter, the specific security requirements of SGs are presented in order to raise awareness of the new security challenges. Finally, a new classification of cyber attacks, based on the architecture of the SG, is proposed and details for each category are provided. The new classification is distinguished by its focus on the cyber-physical security of the SG in particular, which gives a comprehensive overview of the different threats. Thus, this new classification forms the necessary knowledge-basis for the design of respective countermeasures.