Let $\mathcal G$ be a hypergraph whose edges are colored. An {\it $(α,n)$-detachment} of $\mathcal G$ is a hypergraph obtained by splitting a vertex $α$ into $n$ vertices, say $α_1,\dots,α_n$, and sharing the incident edges among the subvertices. A detachment is {\it fair} if the degree of vertices and multiplicity of edges are shared as evenly as possible among the subvertices within the whole hypergraph as well as within each color class. In this paper we solve an open problem from the 1970s by finding necessary and sufficient conditions under which a $k$-edge-colored hypergraph $\cG$ has a fair detachment in which each color class is connected. Previously, this was not even known for the case when $\cG$ is an arbitrary graph (i.e. 2-uniform hypergraph). We exhibit the usefulness of our theorem by proving a variety of new results on hypergraph decompositions (in Part I), and completing partial regular combinatorial structures (in Part II).

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