This master thesis is consisted of two parts. In the first part we study T.-J. Stieltjes' last paper on continued fractions. In this beautiful work, he studied the analytical behavior of the continued fraction of the form 1 a1z + 1 a2+. . . + 1 2k−1z+ 1 a2k+. . . , where a1, a2, . . . , an, . . . are all positive numbers and z is a complex number. Among the other mathematical notions, which he defines, is the famous type of integral which nowadays has his name. In the second part we study some important papers which are related to the history of Stieltjes integral and were published by F. Riesz, H. Lebesgue and J. Radon at the beginning of the 20th century.