Surrogate models are invaluable tools that greatly assist the process of computationally expensive analyses and optimization. Engineering optimization reaps the benefit from surrogate models in order to perform expensive optimization that could potentially be computationally intractable in the pre-high-performance computing age. Moreover, surrogate models provide a means to allow engineering design exploration with high-fidelity computer simulations. Despite their wide use and substantial research progresses, there are still some key issues and challenges that need to be addressed by researchers. Most of these issues stem from the growing complexity of engineering design optimization and exploration in real-world problems. In other words, the sophistication of the problem that we have to tackle increases faster than that of computing power and technology. It is thus imperative to have accurate and yet computationally efficient surrogate models that are suitable for real-world engineering problems. In this paper, we discuss key issues and challenges of the application of surrogate models in engineering design optimization and exploration. This paper is directed toward general readers, in which we aim to present general discussions regarding the effectiveness, issues, and future of surrogate-based optimization and exploration in engineering.