This paper introduces a new term namely fuzzy weighted-k-out-of-n system and studies the fuzzy reliability of two-stage fuzzy weighted-k-out-of-n model having common components following triangular membership function. Methods are developed to find the system fuzzy reliability and the minimal cuts and minimal paths for two-stage fuzzy weighted-k-out-of-n systems have been generated. Fuzzy reliability bounds for systems with s-dependent component failures are also generated. Two types of two-stage fuzzy weighted-k-out-of-n models, the Series Fuzzy Weighted (SFW)-k-out-of-n model, and the Parallel Fuzzy Weighted (PFW)-k-out-of-n model, have been applied to evaluate fuzzy reliability in network applications (project management, and the shortest path problems).