We introduce the exact coloured k -enclosing object problem: given a set P of n points in R 2 , each of which has an associated colour in f 1 ;:::;t g , and a vec- tor c = ( c 1 ;:::;c t ), where c i 2 Z + for each 1 i t , nd a region that contains exactly c i points of P of colour i for each i . We examine the problems of nd- ing exact coloured k -enclosing axis-aligned rectangles, squares, discs, and two-sided dominating regions in a t -coloured point set