The purpose of this work is to demonstrate the utility of differential geometry in modern physics using classical mechanics as an example. The work is of an expository nature and I make no claims of originality (apart from organizing the material). Classical mechanics draws its roots from time of I. Newton (17. cent.), but it was further developed, amongst others, by the likes of J. L. Lagrange (18. cent.) and R. Hamilton (19. cent.). Finally, in the 20th century, after modern differential geometry was sufficiently developed, it became clear that the most convenient language in which to express the culmination of all these ideas was precisely symplectic geometry. To be more specific: symplectic geometry treats the so-called canonical formalism, i.e., Hamiltonian mechanics,which is closely linked to the cotangent bundle(phasespace). Lagrangianmechanics,onthe other hand, is linked to the tangent bundle (position-velocity space) and calculus of variations. Lastly, Newtonian mechanics is captured by Newton’s equations on a Riemannian manifold, i.e., configuration space (whose metric is given by a quadratic form which represents kinetic energy). In this work we will concern ourselves only with the first two formalisms. Although it is not traditional (in the physics community) to approach mechanics this way (primarily because of the mathematical preliminaries), the benefits of doing so are plentiful: notions, which may have only had a strictly analytical meaning, can also be interpreted geometrically, the structure of the theory itself becomes more transparent and the whole discussion can be carried out in a concise and elegant manner. To further emphasize conciseness, most proofs are only sketches in which we skip the routine parts and focus only on the more subtle steps.