This chapter focuses on network flows, specifically the problem of transporting as many units as possible from a source vertex \( s \) to a sink vertex \( t \) in a directed graph \( G \) with edge capacities \( u : E(G) \rightarrow \mathbb{R}_+ \). A flow \( f \) is defined as a function that satisfies the capacity constraints and the flow conservation rule. The value of an \( s \)-flow is defined as \( -\text{ex}_f(s) \), where \( \text{ex}_f(v) \) is the excess of the flow at vertex \( v \). The chapter introduces the basic problem of finding a maximum flow and discusses its applications, such as the job assignment problem. It also outlines a combinatorial algorithm for solving the maximum flow problem using binary search and modeling the problem as an \( s-t \)-flow in a bipartite digraph. Section 8.1 presents the Max-Flow-Min-Cut Theorem, which shows that the maximum flow in a network is equal to the minimum capacity of an \( s-t \)-cut. This theorem is proven using a linear programming (LP) formulation of the maximum flow problem. The chapter also discusses the existence of an optimum integral flow for integral capacities and the implications for Menger's Theorem on disjoint paths. Sections 8.3, 8.4, and 8.5 provide efficient algorithms for the maximum flow problem, while Section 8.6 describes an efficient way to store the minimum capacity of an \( s-t \)-cut for all pairs of vertices. Section 8.7 explores more efficient methods for determining edge-connectivity or a minimum capacity cut in an undirected graph.This chapter focuses on network flows, specifically the problem of transporting as many units as possible from a source vertex \( s \) to a sink vertex \( t \) in a directed graph \( G \) with edge capacities \( u : E(G) \rightarrow \mathbb{R}_+ \). A flow \( f \) is defined as a function that satisfies the capacity constraints and the flow conservation rule. The value of an \( s \)-flow is defined as \( -\text{ex}_f(s) \), where \( \text{ex}_f(v) \) is the excess of the flow at vertex \( v \). The chapter introduces the basic problem of finding a maximum flow and discusses its applications, such as the job assignment problem. It also outlines a combinatorial algorithm for solving the maximum flow problem using binary search and modeling the problem as an \( s-t \)-flow in a bipartite digraph. Section 8.1 presents the Max-Flow-Min-Cut Theorem, which shows that the maximum flow in a network is equal to the minimum capacity of an \( s-t \)-cut. This theorem is proven using a linear programming (LP) formulation of the maximum flow problem. The chapter also discusses the existence of an optimum integral flow for integral capacities and the implications for Menger's Theorem on disjoint paths. Sections 8.3, 8.4, and 8.5 provide efficient algorithms for the maximum flow problem, while Section 8.6 describes an efficient way to store the minimum capacity of an \( s-t \)-cut for all pairs of vertices. Section 8.7 explores more efficient methods for determining edge-connectivity or a minimum capacity cut in an undirected graph.