  The observable universe could be a 1+3-surface (the "brane") embedded in a
1+3+\textit{d}-dimensional spacetime (the "bulk"), with Standard Model
particles and fields trapped on the brane while gravity is free to access the
bulk. At least one of the \textit{d} extra spatial dimensions could be very
large relative to the Planck scale, which lowers the fundamental gravity scale,
possibly even down to the electroweak ($\sim$ TeV) level. This revolutionary
picture arises in the framework of recent developments in M theory. The
1+10-dimensional M theory encompasses the known 1+9-dimensional superstring
theories, and is widely considered to be a promising potential route to quantum
gravity. At low energies, gravity is localized at the brane and general
relativity is recovered, but at high energies gravity "leaks" into the bulk,
behaving in a truly higher-dimensional way. This introduces significant changes
to gravitational dynamics and perturbations, with interesting and potentially
testable implications for high-energy astrophysics, black holes, and cosmology.
Brane-world models offer a phenomenological way to test some of the novel
predictions and corrections to general relativity that are implied by M theory.
This review analyzes the geometry, dynamics and perturbations of simple
brane-world models for cosmology and astrophysics, mainly focusing on warped
5-dimensional brane-worlds based on the Randall--Sundrum models. We also cover
the simplest brane-world models in which 4-dimensional gravity on the brane is
modified at \emph{low} energies -- the 5-dimensional Dvali--Gabadadze--Porrati
models. Then we discuss co-dimension two branes in 6-dimensional models.
