The Conjugate Gradient method is an efficient algorithm for solving linear systems Ax = b where A is symmetric positive definite, and for unconstrained optimization of quadratic functions. Instead of steepest descent directions, it uses conjugate directions that ensure each iteration makes progress independent of previous iterations. For n-dimensional problems, it theoretically converges in at most n iterations (without roundoff errors). In practice, it's used iteratively for large sparse systems. Nonlinear conjugate gradient extends this to general smooth functions, commonly used in large-scale optimization and machine learning.