Thermodynamic formalism as a mathematical theory has its roots in one of the most successful theories of physics -- thermodynamics and statistical mechanics. In its core, the theory of thermodynamic formalism seeks to describe properties of observable "macroscopic" phenomena based on the average behavior of the "microscopic" constituents. In the language of dynamical systems: given a dynamical system $(X, f)$, with $X$ the phase space and $f$ the map defining the dynamics, one seeks to describe properties of functions defined on $X$ (the macroscopic observables) based on the (often averaged, in some well-defined sense) behavior of $f$. In particular, thermodynamic formalism leads to strong results in dimension theory of dynamical systems (e.g. describing fractal dimensions and measures of sets arising as invariant sets of some chaotic dynamical systems). In this first of a series of two talks, we shall present the main ingredients of thermodynamic formalism: topological entropy, metric entropy, topological pressure, and the variational principle for the pressure.