In this chapter, we are going to explore the field of non-linear programming, that is to say, problems consisting in minimising or maximising an objective function subject to a finite number of equality and/or inequality constraints. We begin by introducing the foundational tools required for the analysis of non-linear programming problems, such as feasible, descent and tangent directions. Building on these, we will derive and examine the essential first-order and second-order optimality conditions that form the core of non-linear optimisation theory.