This paper presents two general and efficient methods for determining intersection and self-intersection loci of rationally parametrized surfaces. One of the methods, based on regular systems, is capable of computing the exact parametric loci of intersection and self-intersection. The other, based on Gröbner bases, can compute the minimal varieties passing through the exact parametric loci. The relation between the results computed by the two methods is established and algorithms for computing parametric loci of intersection and self-intersection are described. Experimental results and comparisons with some existing methods show that our algorithms have a good performance.