Title: Efficient Simulation For Tail Probabilities Of Gaussian Random Fields Abstract: We are interested in computing tail probabilities for the maxima of Gaussian random fields. In this paper, we discuss two special cases: random fields defined over a finite number of distinct point and fields with finite Karhunen-Lo\`{e}ve expansions. For the first case we propose an importance sampling estimator which yields asymptotically zero relative error. Moreover, it yields a procedure for sampling the field conditional on it having an excursion above a high level with a complexity that is uniformly bounded as the level increases. In the second case we propose an estimator which is asymptotically optimal. These results serve as a first step analysis of rare-event simulation for Gaussian random fields.