This chapter in detail discusses the gradient descent methods such as the steepest descent method and conjugate gradient method. The methods have been discussed step-wise with detailed illustrative examples. The elaboration on minimization and maximization problems is provided. The chapter touches upon every concept involving these methods, such as necessary condition, sufficiency condition, desirable search direction, descent direction and descent condition. These concepts have been exhibited graphically as well as mathematically. In addition, the important concept involved such as step size determination is evaluated with examples and illustrations.