For most of our history, we've rested easy in the notion that there were three dimensions that have existed throughout time: length, width and height. Ah, the good old days. In the early 20th century, Hermann Minkowski and Albert Einstein connected our comfortable three dimensions with a fourth, time, defining special relativity using a space-time continuum.
This kind of worked, but still didn't explain a troublesome new theory of gravity called quantum mechanics that arose around the same time Minkowski and Einstein were working on their theories. Quantum mechanics had its own rules that contradicted the concepts behind the space-time continuum. Scientists treated this incompatibility like the weather for decades, discussing it but not really doing anything about it.
While higher theoretical dimensions began with Descartes in the 1600s, in the 1970s string theory expanded on this idea as physicists attempted to tie everything together in one elegant explanation of the universe. Variations of string theory require the existence of up to eleven dimensions and a slew of universes, with our universe forming a three-dimensional membrane floating around some higher-dimensional donut. According to this theory, each point in space has six higher dimensions wrapped up in super-tiny geometries called Calibi-Yau Manifolds.
One recent string theory suggests that the reason we only experience the three spatial dimensions is that all universes with higher dimensions got into some cosmic car accident and destroyed each other, leaving our measly three-dimensional brane untouched. List of all episodes here: Through The Wormhole.