The subject of this paper is the interesting topic of solving combinatorial problems using network flows. The paper first defines the basic concepts and presents examples related to the areas of undirected and directed graphs, networks and network flows, as edges, paths, flows and edge capacity. After that, we introduce the central problem of finding the maximum flow in the given network. We prove that the problem is equivalent to the problem of the minimum cut in the given network, so that we come to the most important Max-flow min-cut theorem. After introducing the necessary terms and principles, we will present two commonly used algorithms used to find the maximum flow in the given networks: Algorithms of Ford and Fulkerson and the Dinitz algorithm. Starting from a given flow, both algorithms, increase it until the maximum flow is reached. The Ford-Fulkerson algorithm achieves this by means of augmented paths, while Dinitz algorithm achieves this by means of auxiliary networks and “blocking flows”. As large share of combinatorial problems can be displayed and solved using a simplified version of networks with 0-1 flows, we will analyze the complexity of the Dinitz algorithm for that type of networks. Afterwards we will show and explain several application of network flows in solving selected combinatorial problems: Maximum matching in bipartite graphs and PERT digraph problems. Finally, we use two practical problems of maximum matching in bipartite graphs: Worker distribution and Speed dating, to show how they can be solved using network flows. Here we will rely and test efficiency of two algorithms presented in previous chapters Dinitz algorithms and algorithms of Ford and Fulkerson.