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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 undergo 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 in 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 and scattering time (τ) contributed by the Galactic disk, and independently constructed electron density models for the Galactic halo and other galaxies’ ISMs and halos that account for different galaxy morphologies, masses, densities, and strengths of turbulence. For source redshifts 0.5 ≤z_s≤ 1, an all-sky, isotropic FRB population has simulated values ofτ(1 GHz) ranging from ∼1μs to ∼2 ms (90% confidence, observer frame) that are dominated by host galaxies, althoughτcan be ≫2 ms at low Galactic latitudes. A population atz_s= 5 has 0.01 ≲τ≲ 300 ms at 1 GHz (90% confidence), dominated by intervening galaxies. About 20% of these high-redshift FRBs are predicted to haveτ> 5 ms at 1 GHz (observer frame), and ≳40% of FRBs betweenz_s∼ 0.5–5 haveτ≳ 1 ms forν≤ 800 MHz. Our scattering predictions may be conservative if scattering from circumsource environments is significant, which is possible under specific conditions. The percentage of FRBs selected against from scattering could also be substantially larger than we predict if circumgalactic turbulence causes more small-scale (≪1 au) density fluctuations than observed from nearby halos.