In this thesis, we explore the Riemann-Roch theorem for algebraic curves. We introduced the necessary notions from algebraic geometry, proved the Riemann-Roch theorem, and stated some of its applications. We started with a brief introduction to algebraic geometry, in which we defined affine and projective varieties, and the maps between them. Later we focused specifically on smooth curves, i.e. nonsingular projective varieties of dimension 1. We introduced function fields that form a category, that is equivalent to the category of smooth curves. We demonstrated how statements about function fields and curves can be translated from one context to another. For each curve, we defined the associated divisor group and established some of the basic properties of divisors. To each divisor, there is an associated Riemann-Roch space. It is made up of rational functions with prescribed zeros and poles. We defined the genus of a curve and proved a couple of results about Riemann-Roch spaces. We proved the Riemann inequality, which relates the dimension of a Rimeann-Roch space, the degree of a divisor, and the curve genus. Then, we introduced adele rings and Weil differentials to characterize the index of speciality and prove the Riemann-Roch theorem which improves the Riemann inequality. We stated some applications of the Riemann-Roch theorem: we determined the properties of the canonical divisor, we characterized smooth curves of genus 0 containing a rational point up to isomorphism, we proved that each elliptic curve is isomorphic to a curve defined by Weierstrass equation, we defined the group law on the elliptic curve, and we proved some results about special divisors, including Clifford's theorem.