The growing interest in the use of offshore platforms in deeper waters and harsher environments, as well as the desire to extend the operation of existing structures beyond their design lives, is increasing attention on the assessment of their dynamic response and their failure conditions under extreme storm loading. There are a large number of factors influencing the performance of dynamically sensitive platforms, but a major issue is how to include all of the irregularity, directionality and nonlinearity that ocean waves cause on their loading. Therefore, the role each plays in the assessment of dynamically sensitive structures under extreme loads is investigated systematically in this thesis. The aim is to develop practical methods to estimate extreme response and the probability of failure of dynamically sensitive offshore structures in a given sea-state. The directionality and nonlinearity of ocean waves has been captured in this thesis by extending the formulations of the NewWave and Constrained NewWave theories. NewWave is a deterministic method that accounts for the spectral composition of the sea-state, and can be used as an alternative to both regular wave and full random time domain simulations of lengthy time histories. Based on this theory a predetermined crest height and the surface elevation around the crest during an extreme event can be theoretically simulated. Constrained NewWave, which is generated by mathematically constraining a NewWave within a random time series, allows the irregularity of ocean waves to be considered. These wave theories have been extended in this thesis to include 2nd order and directionality effects, and their formulations have been written into a new Fortran code for calculation of the water surface and water particle kinematics. The effects of irregularity, directionality and nonlinearity of ocean waves on dynamically sensitive structures are then shown for an example mobile jack-up drilling platform. The sample jack-up platform is modelled in the USFOS software, and includes the effects of material and geometrical nonlinearities as well as spudcan-soil-structure nonlinear interactions. Finally, based on structured application of multiple Constrained NewWaves in combination with the Monte Carlo method, a framework is proposed to estimate the extreme response and the failure probability of dynamically sensitive offshore structures exposed to a given duration of the one extreme sea-state. The results demonstrate that in an extreme event the irregularity, directionality and nonlinearity of ocean waves have considerable effects on the overall performance of the sample jack-up platform. It is shown that the extreme response and the probability of failure of the sample jackup are not only governed by the maximum crest elevation but also depend on the random background of ocean waves. In addition, it is indicated that the inclusion of the directionality effects of ocean waves results in reductions in the extreme response and failure rate of the sample jack-up platform. On the other hand, the nonlinearity effects cause additional energy in low and high frequencies and raise the crest height, which increases the extreme response of the sample jack-up platform. The methods developed in this thesis have application to any dynamically sensitive structure and will help reduce the level of uncertainty in predicting their extreme response or failure probability. This may help in extending their operational conditions, say into deeper waters and harsher sea-states, or in extending their operational life.