Radio wave scattering can cause severe reductions in detection sensitivity for surveys of Galactic and extragalactic fast (∼ms duration) transients. While Galactic sources like pulsars are subject to scattering in the Milky Way interstellar medium (ISM), extragalactic fast radio bursts (FRBs) can also experience scattering in their host galaxies and other galaxies intervening their lines-of-sight. We assess Galactic and extragalactic scattering horizons for fast radio transients using a combination of NE2001 to model the dispersion measure (DM) and scattering time (τ ) contributed by the Milky Way, and independently constructed electron density models for other galaxies' ISMs and halos that account for different galaxy morphologies, masses, densities, and strengths of turbulence. For FRB source redshifts z s 1, an all-sky, isotropic FRB population has values of τ ranging between ∼ 1 µs and ∼ 2 ms at 1 GHz (observer frame) that are dominated by host galaxies. For a hypothetical, high-redshift (z s ∼ 5) FRB population, τ ranges from ∼ 0.01 − 100s of ms at 1 GHz, and is largely dominated by intervening galaxies. About 20% of these high-redshift FRBs are predicted to have τ > 5 ms at 1 GHz (observer frame), and 40% of FRBs between z s ∼ 0.5 − 5 are predicted to have τ 1 ms for ν ≤ 800 MHz. The percentage of FRBs selected against from scattering may be substantially larger because our scattering predictions are conservative compared to localized FRBs, and if circumgalactic turbulence causes density fluctuations larger than those observed from nearby halos.