Avoidance of collapse is the most important objective in earthquake resistant design, but assessing the probability of collapse of a structure during an earthquake is technically challenging and it is computationally demanding. This paper summarizes recent investigations conducted by the authors at Stanford University aimed at improving the assessment of the probability of collapse of structures. It is shown that methodologies based on estimating the probability of collapse at a single ground motion intensity, such as those that only evaluate the level of safety at the so called maximum considered earthquake are inadequate. Results are presented showing that estimating the probability of collapse using incremental dynamic analyses in which a model of the structure is subjected to ground motions scaled at increasing levels of intensity until collapse is produced may introduce significant bias in the results. An improved procedure, referred to as Enhanced Two Stripe Analysis, E2SA is presented which provides not only more accurate results compared to those obtained with an incremental dynamic analysis but can be obtained at a small fraction of the computational effort. The new approach is based on careful observation of the deaggregation the mean annual frequency of collapse that reveals that is typically dominated by earthquake ground motion intensities corresponding to the lower half of the collapse fragility curve. Therefore, rather than focusing on the estimation of the median collapse intensity as proposed in previous studies, the new method proposes to focus on intensities in the lower tail of the collapse fragility function. Furthermore, it is shown that the uncertainty in the collapse fragility curve and on the mean annual frequency of collapse is significantly reduced by increasing the number of ground motions used in the analysis, so instead of using a small ground motion set scaled at many increasing levels of intensity, the proposed method recommends conducting nonlinear response history analyses with a larger number of ground motions but only at two levels of intensity. Results indicate that using a larger number of ground motions at two intensity levels provides improved results. Finally, recent investigations conducted by the authors show that rather than selecting and scaling ground motions based on spectral accelerations at a period equal to the fundamental period of the structure alone or in combination with epsilon, a much better approach is to use an averaged spectral acceleration over a wide range of periods extending to periods shorter than the fundamental period to significantly longer than the fundamental period. Extensive studies conducted by the authors over a wide range of structures indicate that this new intensity measure is significantly better correlated with collapse and therefore leads to a more reliable estimation of the mean annual frequency of collapse while at the same time reducing the computational effort involved since the number of ground motions to be used in the analysis can be reduced.