This chapter provides an introduction to key concepts in optimization with motivating applications. We begin with a categorization of different optimization problems, based on the objective function, to be maximized or minimized, and constraint equations that limit the set of feasible solutions. Next we introduce two major concepts in optimization: convexity and the gradient. Convex problems are essentially solved, at least numerically. Linear programming and quadratic programming are both convex. Nonconvex problems are more challenging. However, there is a shift in optimization, where it is increasingly possible to solve nonconvex problems. The gradient is another fundamental concept, providing information about the geometry of the optimization landscape, and which direction locally improves the solution. We then show how Lagrange multipliers are an intuitive way of handling constraints. We conclude with three major applications: inverse problems, control theory, and machine learning. Throughout, we give examples to illustrate how to formulate an optimization problem and solve it algorithmically.