This course is intended to serve as a first introduction to stochastic processes, with an emphasis on theoretical aspects and proofs, without requiring full measure theoretical foundations, but only requiring a background course in basic probability theory such as Math 301. The course provides the students with an introduction to examples of stochastic processes much sought for in the industry, such as discrete and continuous time Markov chains, martingales and Brownian motion. In addition, connection with other areas of mathematics such as the amenability problem in group theory and various applications in random geometry will be discussed. The course will equip students intending to continue in diverse pure or applied areas such as group theory, random geometry, financial mathematics, dynamical systems, probabilistic aspects of data science or telecommunication networks, with essentials of the subject.