As has already been mentioned when we discussed the previous chapter, in the current one we introduce and thoroughly study the objects related to the powerful concept of nice sets, which will be our indispensable tool in the last part of the second volume of the book, leading, along with the theory of conformal graph directed Markov systems, the first return map method, and the techniques of Young towers, to such stochastic laws as the exponential decay of correlations, the Central Limit Theorem, and the Law of the Iterated Logarithm follows for large classes of elliptic functions constituted by subexpanding and parabolic ones. However, the main objective of this chapter is to prove the existence of nice and pre-nice sets.