This chapter focuses on nonlinear fractional programming problems, specifically the problem: \[ P: \max_{x \in S} q(x) = \frac{f(x) + \alpha}{g(x) + \beta} \] where \( f, g: R^n \to R \), \(\alpha, \beta \in R\), and \( S \subseteq R^n \). The assumptions are that \( g(x) + \beta > 0 \) for all \( x \in S \), \( S \) is non-empty, and the objective function has a finite optimal value. The methods for solving this problem are categorized into three types: 1. **Methods based on the change of variables**: Transform the problem \( P \) into a simpler problem \( P' \). 2. **Direct methods**: View \( P \) as a nonlinear programming problem and use known methods. If \( f \) and \( g \) satisfy certain convexity or concavity hypotheses, the function \( q \) has properties that facilitate finding the optimal solution, such as pseudoconcavity. 3. **Parametric methods**: Associate the problem with a parametric problem and solve it for various values of the parameter. The chapter also discusses necessary and sufficient optimality conditions for the nonlinear fractional programming problem with nonlinear constraints. The conditions involve the existence of \( m \) scalars \( u_i \) that satisfy specific equations, ensuring that \( x_0 \) is an optimal solution. The proof of these conditions is provided, focusing on the sufficiency part.This chapter focuses on nonlinear fractional programming problems, specifically the problem: \[ P: \max_{x \in S} q(x) = \frac{f(x) + \alpha}{g(x) + \beta} \] where \( f, g: R^n \to R \), \(\alpha, \beta \in R\), and \( S \subseteq R^n \). The assumptions are that \( g(x) + \beta > 0 \) for all \( x \in S \), \( S \) is non-empty, and the objective function has a finite optimal value. The methods for solving this problem are categorized into three types: 1. **Methods based on the change of variables**: Transform the problem \( P \) into a simpler problem \( P' \). 2. **Direct methods**: View \( P \) as a nonlinear programming problem and use known methods. If \( f \) and \( g \) satisfy certain convexity or concavity hypotheses, the function \( q \) has properties that facilitate finding the optimal solution, such as pseudoconcavity. 3. **Parametric methods**: Associate the problem with a parametric problem and solve it for various values of the parameter. The chapter also discusses necessary and sufficient optimality conditions for the nonlinear fractional programming problem with nonlinear constraints. The conditions involve the existence of \( m \) scalars \( u_i \) that satisfy specific equations, ensuring that \( x_0 \) is an optimal solution. The proof of these conditions is provided, focusing on the sufficiency part.