This chapter is a survey of the main approximation schemes for Quantum Mechanics. We begin from the semiclassical methods. After some generalities, we study the asymptotic limit of path integrals for small \(\hbar \) , compute the Van Vleck determinant, and explain its relation to the Maslov index. We then develop the semi-classical instanton calculus and derive the dilute gas approach which is fully justified. We describe the semi-classical approximation for the Schrödinger equation, and construct the Riccati recursion. We conclude the review of semi-classical methods with the Bohr-Sommerfeld quantization. We illustrate the large-N methods focusing on the \(O(N)\) model. We examine three different approaches to perturbation theory: (1) time-independent Hilbert-space methods; (2) time-dependent interaction-picture techniques; and (3) path integral perturbative expansions. Using the path integral framework, we describe the graphical Feynman rules for: (a) vacuum correlation functions; (b) connected correlations; and (c) the generating functionals. We prove the quantum adiabatic theorem, and outline the adiabatic methods; in this context we introduce the Berry phase, and elaborate on its geometrical and physical meanings. We finally consider the Born-Openhamer approximation.