**Abstract:** Hurwitz spaces, introduced at the end of 19th century, consist of classes of isomorphism of pairs (algebraic curve, rational function on it), where a curve has a fixed genus and a function has a fixed degree. The strata of the filtration, to which the talk is devoted, are formed by the pairs, in which a function has a fixed number of \textit{critical values}. In every Hurwitz space the largest stratum (the \textit{Morse} one) is Zariski-open, while the lowest one consists of pairs in which the function has only three critical values, i.e. of \textit{Belyi pairs}. The considerable part of the talk will be devoted to the strata closest to the lowest ones, i.e. the so-called \textit{Fried families}. The combinatorial, algebro-geometric and arithmetic structures, related to these objects, will be considered, and some examples will be presented.