Excursion probabilities of Gaussian random fields have many applications in statistics (e.g., scanning statistic and control of false discovery rate (FDR)) and in other areas. The study of excursion probabilities has had a long history and is closely related to geometry of Gaussian random fields. In recent years, important developments have been made in both probability and statistics. In this talk, we consider the excursion probabilities of two types of bivariate Gaussian random fields: those with non-smooth (or fractal) sample functions, and those with smooth sample functions. An important class of stationary multivariate Gaussian random fields with prescribed smoothness properties are those with Mat\'ern cross-covariance functions studied by Gneiting, Kleiber, and Schlather (2010). (i) For not necessarily smooth bivariate Gaussian random fields, we apply the double sum method to prove precise asymptotic results for the excursion probability, which refine and extend those of Pickands (1969), Piterbarg (1996) and Piterbarg and Stamatovich (2005). (ii) For smooth stationary bivariate Gaussian random fields, we show that the results in (i) can be significantly strengthened and the ``Expected Euler Characteristic Heuristic'' still holds. The methods for establishing these two types of results are very different.