  Teleparallel gravity has significantly increased in popularity in recent
decades, bringing attention to Einstein's other theory of gravity. In this
Review, we relate this form of geometry to the broader metric-affine approach
to forming gravitational theories where we describe a systematic way of
constructing consistent teleparallel theories that respect certain physical
conditions such as local Lorentz invariance. We first use teleparallel gravity
to formulate a teleparallel equivalent of general relativity which is
dynamically equivalent to general relativity but which may have different
behaviors for other scenarios, such as quantum gravity. After setting this
foundation, we describe the plethora of modified teleparallel theories of
gravity that have been proposed in the literature. In the second part of the
Review, we first survey works in teleparallel astrophysics literature where we
focus on the open questions in this regime of physics. We then discuss the
cosmological consequences for the various formulations of teleparallel gravity.
We do this at background level by exploring works using various approaches
ranging from dynamical systems to Noether symmetries, and more. Naturally, we
then discuss perturbation theory, firstly by giving a concise approach in which
this can be applied in teleparallel gravity theories and then apply it to a
number of important theories in the literature. Finally, we examine works in
observational and precision cosmology across the plethora of proposal theories.
This is done using some of the latest observations and is used to tackle
cosmological tensions which may be alleviated in teleparallel cosmology. We
also introduce a number of recent works in the application of machine learning
to gravity, we do this through deep learning and Gaussian processes, together
with discussions about other approaches in the literature.
