Do we know how many dimensions there are?
Physicists claim to have a beautiful and consistent theory of quantum gravity, yet they can't agree on the number of dimensions. Some say 10, and some say 11. Actually, our universe may have both 10 and 11 dimensions.
The classical notion of space-time loses its meaning in quantum theory. It's replaced by a more general concept, a "quantum space-time" where dimension is not a well-defined notion. We know a great deal about certain quantum space-times, which have many precise properties, but dimension is not one of those properties.