We propose an integrated scheme for transient stability assessment which in a sequence screens contingencies and scrutinizes only the selected ones. This scheme is based on a hybrid method, called SIME for SIngle Machine Equivalent. SIME relies on a particular direct method coupled with time-domain programs so as to combine the strengths of both, namely: the flexibility with respect to power system modelling of time-domain methods; the speed and richer information of the direct method. This paper lays the foundations of SIME, devises appropriate techniques for transient stability assessment per se and for contingency screening, and finally integrates these two techniques in a fully general function, i.e. able to comply with any power system modelling and stability scenario, and to assess any type of stability limits (critical clearing times or power limits). Throughout, real-world examples illustrate the proposed techniques and highlight their performances.