We will discuss several computational and theoretical aspects of computing the rank of Nèron-Severi groups of varieties over finite fields and number fields. In particular, we will explore the rank estimates for elliptic surfaces, which are interesting from arithmetic point of view. Recent development in this area allows some explicit numerical computations. Most of the work is based on the results of T. Shioda, R. Kloosterman, R. van Luijk and B. Poonen. We will also investigate the role and application of Tate and Artin-Tate conjectures. For K3 surfaces those conjectures are proved and can be applied to get sharp upper bounds for the rank of the Nèron-Severi group.