In the paper, approaches to constructing test sequences are considered in the case where only admissible input actions on the system are known, whereas no information about the states of the system or transitions between them in response to these actions is available. Two approaches to constructing test sequences that guarantee the widest variety of situations arising in the course of testing are suggested. The first approach gives rise to the so-called de Bruijn sequences. The second approach yields sequences covering all states or transitions in all finite automata with the number of states not exceeding a given constant. Both kinds of sequences are related to the combinatorial analysis of words in finite alphabets. Some methods for constructing such sequences are also discussed.