In this paper we talk about the network flow problem. First of all, we define some of the basic terms needed for the future analysis of the problem. Then we describe the Ford-Fulkerson algorithm, which is one of the most important algorithms for solving the network flow problem. We analyse the complexity of the Ford-Fulkerson algorithm and also demonstrate by example how to compute the maximum flow by using this algorithm. Before the complexity analysis, we first talk about Max-Flow/Min-Cut Theorem, that talks about the relationship between maximum flow and minimum cut problems. After describing Ford-Fulkerson’s algorithm, we describe the Edmonds-Karp algorithm, which is, actually, one of improvements of the Ford-Fulkerson algorithm. Finally, we introduce some extensions of the maximum flow problem, like circulations with demands and circulations with demands and lower bounds. Above mentioned extensions of the maximum flow will be used to solve the problem of survey design and airline scheduling. Survey design and airline scheduling are just some applications of the maximum flow problem.