This paper sets out the main features of the constructivist school, as promoted by Andrej Andreevich Markov (1903–1979). After a short survey of the situation pertaining in mathematics and logic, at the beginning of the 20th century, the emergence of intuitionism, and of recursive function theory, is sketched in. The paper then outlines the aims and methods of Markov's constructivism — reviewing, by way of illustration, the major results obtained through Markov's approach, in the field of real-number analysis. Finally, emphasis is laid on the current relevance of such results for present-day mathematics.