We consider certain pathological point sets from the general measure theoretical point of view. Namely, we discuss Mazurkiewicz sets, also called two-point sets, which have difficult and interesting descriptive as well as measure theoretic properties. Moreover, we will discuss also uniform subsets of the Euclidean space and their connections to the Mazurkiewizc sets. Also, in the talk will be considered Bernstein sets and Hamel bases.