This book addresses the important topic of Riemann-Stieltjes Inte- gral. First, we study the (apparently) different definitions of Riemann- Stieltjes integral and, in particular, we concentrate on the epsilon-delta definition. We shall use it to study and establish most properties of Riemann-Stieltjes integral. Then, among other things, we will show that the different definitions are equivalent. Second, we study some classes of Riemann-Stieltjes integrable functions, we establish the Fundamental Theorem of Calculus for Riemann-Stieltjes integral, and we, finally, present some methods of Riemann-Stieltjes in- tegration.