This paper is an extended abstract of the invited talk given by the first-named author. In the first part of the talk we give a general introduction to collusion-resistant fingerprinting, discussing problem statements and different sets of assumptions for the digital fingerprinting problem. In the second part we discuss in more detail a combinatorial version of the fingerprinting problem, known as parent-identifying codes. Most earlier works on digital fingerprinting rely on the so-called marking assumption, under which the attackers cannot modify the coordinates in which their fingerprints are identical. We introduce a version of parent-identifying codes for collusion attacks that do not necessarily follow the marking assumption. We show existence of such codes for coalitions of arbitrary size t. Some exact answers are obtained for t=2 pirates.