These lectures contain the material presente'd in a course given at the Tata Institute during the period December 1964 - February J965. The purpose of these lectures was to give an introduction to Grothendieck's theory of the fundamental group in algebraic geometry with, as application, the study of the fundamental group of an algebraic curve over an algebraically closed field of arbitrary characteristic. All of the material (and much more) can be found in the "Seminaire de geometrie algebrique" of Grothendieck, 1960-1961 Expose V, IX and X.