An abstraction of the basic structural pattern underlying any attempt at quantising a physical model yields a functor from a geometric category modelling the spacetime propagation and interactions of the physical entities (particles, strings etc.) into the algebraic category of vector spaces (possibly with additional structure). This very general observation may - under favourable circumstances - lead to highly nontrivial insights and concrete computational results concerning the physical theory and the ambient geometry itself. Emblematic of this line of thought is the development of the Topological Quantum Field Theory, having its origin in the pioneering works of Segal, Witten, Atiyah, Turaev et al., and spanning a remarkable wealth of topics and ideas - from topological invariants generalising the Jones polynomial, all the way to the categorial quantisation programme for two-dimensional Conformal Field Theory and the state-sum models of Quantum Gravity. In these lectures, we present an elementary introduction to the axiomatics of TQFT (Part I), discuss its applications to the study of low-dimensional topology (Part II), and - time permitting (Part III?) - indicate its field-theoretic realisations and generalisations furnished by higher geometric structures that define lagrangean models of dynamics of topologically charged objects.