In a few words, there are theories with extra dimensions and that explain why the universe is expanding as fast as we see it and that propose some dark matter candidates. The number of spatial dimensions is in this case larger than 3.

One way to see this is the following. The extra dimensions can be seen as hidden from us instead of being different from us. Think about a pencil. This is a 3D object. Now, ask a friend to hold the pencil for you and start moving away from it. You will see your 3D pencil appearing like a straight line (1D). That's basically what is going on with the extra dimensions.

Other setups are possible, with large (instead of compact) extra dimensions, but they seem less favored by current data. Moreover, this would not help to explain with his going on in our 3D world.

I hope this helps :)