The cobordism invariants, genera, play fundamental roles in manifold topology. Some specific genera, like A-hat genus, L-genus, Witten genus and elliptic genus are the topological pillars of the Atiyah-Singer index theory, spin geometry and string geometry. In this minicourse, we will give a brief introduction to genera starting from very basic knowledge and introduce some applications of genera in physics in particular anomaly cancellation problem, and in geometry concerning group actions and curvatures.