Efficient global optimization deals with problems in which budget is limited and functions have non-negligible evaluation costs. A successful workaround to this sort of problems is to keep a cheap-to-evaluate surrogate model of the expensive target function which undertakes a major part of function-evaluation during the search process. Gaussian Process Model has been amongst the most widely used surrogate models. It has advantages such as a) providing standard error for its predictions, b) simplicity of definition and inference. However, in its original form, it is afflicted with some shortcomings e.g. cubic time complexity of learning and inference, under-estimated predictive variance and unreasonable presumption about the target values on distant spots from the current observations. The subject of this thesis is to investigate solutions for surrogate-based Efficient Global Optimization under Bayesian framework which overcome the downsides of Gaussian Processes while including its advantages.