The central idea is that the visible, four-dimensional spacetime is restricted to a brane inside a higher-dimensional space, called the "bulk" (also known as "hyperspace").
Some versions of brane cosmology, based on the large extra dimension idea, can explain the weakness of gravity relative to the other fundamental forces of nature, thus solving the hierarchy problem.
[1] Extensions of the large extra dimension idea with supersymmetry in the bulk appear to be promising in addressing the so-called cosmological constant problem.
In 1998/99, Merab Gogberashvili published on arXiv a number of articles where he showed that if the Universe is considered as a thin shell (a mathematical synonym for "brane") expanding in 5-dimensional space then there is a possibility to obtain one scale for particle theory corresponding to the 5-dimensional cosmological constant and Universe thickness, and thus to solve the hierarchy problem.
The ekpyrotic theory hypothesizes that the origin of the observable universe occurred when two parallel branes collided.
[11] The recent multi-messenger gravitational wave event GW170817 has also been used to put weak limits on large extra dimensions.