The latest compulation of string theory and supergravity that brings together a multi-dimensional world view with the Standard Model is called brane world. In the brane world scenario, the entire Universe is an eleven dimensional ‘bulk’ composed of an infinite number of ten dimensional branes. Each brane is consists of a macroscopic 4D spacetime and a compacted, microscopic 6D quantum world (a Calabi-Yau space). The Calabi-Yau sector is twisted beyond all possible detection and houses all the symmetries of the Standard model.
The attractive aspect to the brane world scenario is that three of the four fundamental forces (strong, weak, electromagnetism and their assoicated particles) are represented by open strings. Gravity (and gravitons) are represented by closed strings (loops). Open strings are attached to their respective branes, but closed strings (gravity) are free to move between branes. This explains why gravity is so much weaker than the other forces, and gravity can be used to commiuncate between the branes (gravity phones). This also leads to a natural explanation for dark matter.
All the branes are embedded in a higher dimension all the ‘bulk’, a pile of parallel universes. Thus, the total Universe is eleven dimensional, 4D spacetime + 6D quantum space + 1D bulk. An infinite number of parallel universes is literally just a millimeter away, but outside your 4D vision.