We study the open subset of the Hurwitz space, consisting of meromorphic functions of a given degree defined on closed Riemann surfaces of a given genus and having no real critical values, and enumerate its connected components in terms of braids. Specifically, to a function in this open set, we assign a braid in the braid group of the underlying closed surface and characterize all braids which might appear using this construction. We introduce the equivalence relation among these braids such that the braids corresponding to the meromorphic functions from the same connected component of the above Hurwitz space are equivalent while non-equivalent braids correspond to distinct connected components. Several special families of meromorphic functions, some applications, and further problems are discussed.