The chapter introduces the conjugate gradient method, a powerful technique for solving numerical problems involving large systems with many variables. The method is particularly useful when the coefficient matrix \( A \) is symmetric but indefinite, a common scenario in applications such as fluid dynamics and optimization problems with equality constraints. The chapter discusses the challenges of applying the conjugate gradient method to indefinite systems, including the risk of division by zero and error growth. It reviews various modifications and alternative methods proposed to address these issues, focusing on those that do not break down and can be implemented efficiently. Two main approaches are highlighted: one based on minimizing the sum of squares of residuals and another derived from the biconjugate gradient algorithm. The chapter also reviews recent work by Paige and Saunders, which introduces a method based on the orthogonal reduction of a tridiagonal matrix to an upper triangular form. While this method is complex, it is shown to be equivalent to the second method derived from the biconjugate gradient algorithm, offering more convenient recurrence relations. The chapter concludes by discussing other potential methods for indefinite systems, though they are not as promising as the two methods derived from biconjugate gradients.The chapter introduces the conjugate gradient method, a powerful technique for solving numerical problems involving large systems with many variables. The method is particularly useful when the coefficient matrix \( A \) is symmetric but indefinite, a common scenario in applications such as fluid dynamics and optimization problems with equality constraints. The chapter discusses the challenges of applying the conjugate gradient method to indefinite systems, including the risk of division by zero and error growth. It reviews various modifications and alternative methods proposed to address these issues, focusing on those that do not break down and can be implemented efficiently. Two main approaches are highlighted: one based on minimizing the sum of squares of residuals and another derived from the biconjugate gradient algorithm. The chapter also reviews recent work by Paige and Saunders, which introduces a method based on the orthogonal reduction of a tridiagonal matrix to an upper triangular form. While this method is complex, it is shown to be equivalent to the second method derived from the biconjugate gradient algorithm, offering more convenient recurrence relations. The chapter concludes by discussing other potential methods for indefinite systems, though they are not as promising as the two methods derived from biconjugate gradients.