Within the framework of complex system analyses, such as aircraft and launch vehicles, the presence of computationally intensive models (e.g., finite element models and multidisciplinary analyses) coupled with the dependence on discrete and unordered technological design choices results in challenging modeling problems. In this paper, the use of Gaussian process surrogate modeling of mixed continuous/discrete functions and the associated challenges are extensively discussed. A unifying formalism is proposed in order to facilitate the description and comparison between the existing covariance kernels allowing to adapt Gaussian processes to the presence of discrete unordered variables. Furthermore, the modeling performances of these various kernels are tested and compared on a set of analytical and aerospace engineering design related benchmarks with different characteristics and parameterizations. Eventually, general tendencies and recommendations for such types of modeling problem using Gaussian process are highlighted.