The paper is based on research carried out on secondary school students and students commencing their university studies in mathematics. The basic purpose of the research was to investigate the understanding of the concept of a sequence and to determine the sources of the formation of revealed conceptions. To achieve the objectives an expanded set of selected mathematical situations--simple but not quite standard--were investigated and various other research instruments were used. The students' conceptions were divided into two groups. In the first group a sequence was perceived as a function, in the second it was associated with ordered elements. The diversity of the last set of conceptions was particularly interesting. The students understood the word "ordered" as some kind of relationship between the terms of a sequence, a certain regularity or harmony. In the paper some ways of correcting the conceptions revealed and introducing the concept of a sequence in schools were also discussed.