A partition (C1,C2,...,Cq) of G = (V,E) into clusters of strong (respectively, weak) diameter d, such that the supergraph obtained by contracting each Ci is ℓ-colorable is called a strong (resp., weak) (d, ℓ)-network-decomposition. Network-decompositions were introduced in a seminal paper by Awerbuch, Goldberg, Luby and Plotkin in 1989. Awerbuch et al. showed that strong network-decompositions can be computed in distributed deterministic time. Even more importantly, they demonstrated that network-decompositions can be used for a great variety of applications in the message-passing model of distributed computing. Much more recently Barenboim (2012) devised a distributed randomized constant-time algorithm for computing strong network decompositions with d = O(1). However, the parameter ℓ in his result is O(n1/2 + ε). In this paper we drastically improve the result of Barenboim and devise a distributed randomized constant-time algorithm for computing strong (O(1), O(nε))-network-decompositions. As a corollary we derive a constant-time randomized O(nε)-approximation algorithm for the distributed minimum coloring problem. This improves the best previously-known O(n1/2 + ε) approximation guarantee. We also derive other improved distributed algorithms for a variety of problems. Most notably, for the extremely well-studied distributed minimum dominating set problem currently there is no known deterministic polylogarithmic -time algorithm. We devise a deterministic polylogarithmic-time approximation algorithm for this problem, addressing an open problem of Lenzen and Wattenhofer (2010).