On [a, b]: density 1/(b-a), mean (a+b)/2, variance (b-a)^2 / 12.
That variance formula is worth memorising - the 12 appears constantly and candidates rarely have it ready.
The standard uniform on [0,1] is the base for simulation: by inverse transform sampling, applying the inverse CDF of any distribution to a uniform draw produces that distribution.
Common interview use. Break points on a stick, arrival times within an interval, and order statistics questions where the answer is k/(n+1) by symmetry.