This is the first of two chapters dedicated to the Schrödinger equation for non-relativistic quantum mechanical systems. We start from general properties of the equation: the continuity equation for the probability density, its relation with the classical Hamilton-Jacobi theory, the dichotomy between bound and scattering states, and separability. We then discuss in detail two basic examples: the free particle moving in \(\mathrm {r}^n\) , and the harmonic oscillator which is presented from six different perspectives, including the Lie-theoretic one, Bargmann quantization, and the Mehler formula. Then we discuss the mathematics of one-dimensional Schródinger equations: the Sturm-Liouville theory, the Prüfer form of the Schrödinger equation, and Fuchsian ODEs. Then we specialize our analysis to the several one-dimensional geometries: the line, the half-line, the segment, and the circle. In this context we describe the image method, the Bohm-Aharonov effect, and the band structure of the spectrum for periodic potentials. We conclude the chapter with a survey of a half dozen important examples to highlight crucial physical phenomena as the tunneling effect, the reflection over the barrier, etc. The appendix explains the “SUSY trick” to solve some Schrödinger equations.