We show that each of the Banach spaces C0(ℝ) and Lp(ℝ), 2 < p < ∞, contains a function whose integer translates are complete. This function can also be chosen so that one of the following additional conditions hold: (1) Its non-negative integer translates are already complete. (2) Its integer translates form an orthonormal system in L2(ℝ). (3) Its integer translates form a minimal system. A similar result holds for the corresponding Sobolev space, for certain weighted L2 spaces, and in the multivariate setting. We also prove some results in the opposite direction.