This thesis is dedicated to the study of the cosmological perturbations in the inflationary universe. With the homogenous Robertson-Walker metric as our starting point, we introduce the primordial metric fluctuations which are classified as scalar, vector or tensor. Then, we introduce the gauge invariant variables and study the homogenous background of a universe filled with a scalar field. We also imply the “slow roll approximation” in order to have a smooth and realistic inflationary period. After completing the necessary treatment of the background, we begin studying the perturbations. At first, we link the perturbations of the inflation field with the metric perturbations through the Einstein-Hilbert action and derive the equations of motion. Then, we quantize the inflation field through the canonical quantization and derive the initial conditions that it satisfies. In the final part of the thesis, we derive the power spectrum of the cosmological perturbations. We also link the theoretical predictions of inflation with observable quantities, making a quick introduction to the primordial gravitational waves. Finally, we refer to the related future experiments and the results we expect to obtain.