This chapter deals with conformal graph directed Markov systems, its special case of iterated function systems, and thermodynamic formalism of countable alphabet subshifts of finite type, frequently also called topological Markov chains. This theory started in the mid-1990s with the papers and a book by the second named author and Mauldin. It was there where the concept of conformal measures due to Patterson and Sullivan was adapted to the realm of conformal graph directed Markov systems and iterated function systems. We present here some elements of this theory, primarily those related to conformal measures and Bowen's Formula for the Hausdorff dimension of limit sets of such systems. In particular, we get a cost-free, effective, lower estimate for the Hausdorff dimension of such limit sets. More about conformal graph directed Markov systems can be found in many papers and books. In the second volume of the book, we apply these techniques, by means of nice sets in the next chapter, to get a good, explicit estimate from below of Hausdorff dimensions of Julia sets of elliptic functions and to explore stochastic properties of invariant versions of conformal measures for parabolic and subexpanding elliptic functions.