  We give a pedagogical introduction to time-independent scattering theory in
one dimension focusing on the basic properties and recent applications of
transfer matrices. In particular, we begin surveying some basic notions of
potential scattering such as transfer matrix and its analyticity,
multi-delta-function and locally periodic potentials, Jost solutions, spectral
singularities and their time-reversal, and unidirectional reflectionlessness
and invisibility. We then offer a simple derivation of the Lippmann-Schwinger
equation and Born series, and discuss the Born approximation. Next, we outline
a recently developed dynamical formulation of time-independent scattering
theory in one dimension. This formulation relates the transfer matrix and
therefore the solution of the scattering problem for a given potential to the
solution of the time-dependent Schr\"odinger equation for an effective
non-unitary two-level quantum system. We provide a self-contained treatment of
this formulation and some of its most important applications. Specifically, we
use it to devise a powerful alternative to the Born series and Born
approximation, derive dynamical equations for the reflection and transmission
amplitudes, discuss their application in constructing exact tunable
unidirectionally invisible potentials, and use them to provide an exact
solution for single-mode inverse scattering problems. The latter, which has
important applications in designing optical devices with a variety of
functionalities, amounts to providing an explicit construction for a
finite-range complex potential whose reflection and transmission amplitudes
take arbitrary prescribed values at any given wavenumber.
