A natural way for finding the optimum of a function is searching, which should follow a certain methodic procedure. A first series of derivative-free methods for one-dimensional problems is introduced: dichotomy, interval-halving, Fibonacci, golden series, and interpolations. Continuing in one dimension, derivative-based methods are considered: Newton-Raphson and secant. Next, basic multidimensional searching is explored: steepest ascent and Fletcher-Reeves (conjugate gradients) method. Keeping the multidimensional context, second-order methods based on Hessian are studied, beginning with the Newton’s method, and then the Levenberg-Marquardt method, and then more methods that approximate the Hessian: Broyden, Davidon-Fletcher-Powell, and the BFGS method. An alternative also considered is trust-based methods. The chapter ends with the Gauss-Newton method for vector functions. Programs in MATLAB were included for most of the mentioned methods.