Submodular functions and convexity, by L. Lovász. Convex functions play a central role in continuous optimization, with many applications in economics, engineering, and other sciences. Convexity is a natural property of functions and domains, preserved under various operations, and allows for the development of elegant theories and efficient algorithms. Similarly, submodular set-functions play a key role in discrete optimization. Although submodular functions lack the geometric intuition of convex functions, they have been studied extensively, particularly in the context of matroid rank functions. Edmonds (1970) initiated the systematic study of submodularity, while Choquet (1955) introduced submodular set-functions through a different approach. Submodular functions share several properties with convex functions, including the ability to develop a beautiful and useful theory, and the existence of efficient algorithms for finding minima. Submodular functions are also similar to concave functions, suggesting that maximization problems may yield interesting results. However, maximization is generally more difficult than minimization, and no general solution exists, though special cases and heuristics can be found. Submodular functions give rise to polyhedra with nice properties, and linear programming problems associated with them can be solved using greedy algorithms. For two submodular functions, polyhedral considerations lead to deep min-max results, such as the Matroid Intersection Theorem. Submodularity has been used to generalize classical graph-theoretical results, such as the Marriage Theorem and the Max-flow-min-cut Theorem. These generalizations often remain valid and can be proven using similar techniques. Submodular functions arising from graphs and other combinatorial structures yield surprising results, such as the Matroid Intersection Theorem, which implies Menger's Theorem and the Disjoint Spanning Tree Theorem. This paper surveys the main aspects of submodularity, focusing on fundamental ideas and constructions. It assumes familiarity with matroid theory and recommends further reading in Schrijver's forthcoming book.Submodular functions and convexity, by L. Lovász. Convex functions play a central role in continuous optimization, with many applications in economics, engineering, and other sciences. Convexity is a natural property of functions and domains, preserved under various operations, and allows for the development of elegant theories and efficient algorithms. Similarly, submodular set-functions play a key role in discrete optimization. Although submodular functions lack the geometric intuition of convex functions, they have been studied extensively, particularly in the context of matroid rank functions. Edmonds (1970) initiated the systematic study of submodularity, while Choquet (1955) introduced submodular set-functions through a different approach. Submodular functions share several properties with convex functions, including the ability to develop a beautiful and useful theory, and the existence of efficient algorithms for finding minima. Submodular functions are also similar to concave functions, suggesting that maximization problems may yield interesting results. However, maximization is generally more difficult than minimization, and no general solution exists, though special cases and heuristics can be found. Submodular functions give rise to polyhedra with nice properties, and linear programming problems associated with them can be solved using greedy algorithms. For two submodular functions, polyhedral considerations lead to deep min-max results, such as the Matroid Intersection Theorem. Submodularity has been used to generalize classical graph-theoretical results, such as the Marriage Theorem and the Max-flow-min-cut Theorem. These generalizations often remain valid and can be proven using similar techniques. Submodular functions arising from graphs and other combinatorial structures yield surprising results, such as the Matroid Intersection Theorem, which implies Menger's Theorem and the Disjoint Spanning Tree Theorem. This paper surveys the main aspects of submodularity, focusing on fundamental ideas and constructions. It assumes familiarity with matroid theory and recommends further reading in Schrijver's forthcoming book.