Critical percolation is fairly well-understood on Z^d for d > 11. Exact values of many critical exponents are rigorously known: for instance, the “one-arm” probability that the origin is connected by an open path to distance r scales as r^{-2}. However, most existing methods rely heavily on the symmetries of the lattice, so they do not extend to fractional spaces. We will discuss progress on these questions in the high-dimensional upper half-space, including a proof that the half-space one-arm probability scales as r^{-3}.