I will consider non-local Schrödinger operators obtained as suitable sums of a pseudo-differential (such as a fractional Laplace) operator describing the kinetic term, and a multiplication operator called potential. Two classes of potentials will be considered, those that increase to infinity (“confining”) and those that decrease to zero (“decaying”) at infinity. First I will present some examples when the eigenvalue equation of a non-local Schrödinger operator can be solved explicitly. Then, coming back to general cases and assuming that a ground state for these operators exists, I will discuss a line of improving regularity from slow spatial decay of ground states to intrinsic ultracontractivity. If time permits, I will illustrate these results by presenting some of their implications in relativistic quantum theory.