We consider shot noise processes (X(t))t≥0 with deterministic response function h and the shots occurring at the renewal epochs 0=S0<S1<S2⋯ of a zero-delayed renewal process. We prove convergence of the finite-dimensional distributions of (X(ut))u≥0 as t→∞ in different regimes. If the response function h is directly Riemann integrable, then the finite-dimensional distributions of (X(ut))u≥0 converge weakly as t→∞. Neither scaling nor centering are needed in this case. If the response function is eventually decreasing, non-integrable with an integrable power, then, after suitable shifting, the finite-dimensional distributions of the process converge. Again, no scaling is needed. In both cases, the limit is identified. If the distribution of S1 is in the domain of attraction of an α-stable law and the response function is regularly varying at ∞ with index −β (with 0≤β<1/α or 0≤β≤α, depending on whether ES1<∞ or ES1=∞), then scaling is needed to obtain weak convergence of the finite-dimensional distributions of (X(ut))u≥0. The limits are fractionally integrated stable Lévy motions if ES1<∞ and fractionally integrated inverse stable subordinators if ES1=∞.