In this work, we experimentally investigate the impact of water droplets onto soft viscoelastic surfaces with a wide range of impact velocities. Several impact phenomena, which depend on the dynamic interaction between the droplets and viscoelastic surfaces, have been identified and analyzed. At low $\mathrm{We}$, complete rebound is observed when the impact velocity is between a lower and an upper threshold, beyond which droplets are deposited on the surface after impact. At intermediate $\mathrm{We}$, entrapment of an air bubble inside the impinging droplets is found on soft surfaces, while a bubble entrapment on the surface is observed on rigid surfaces. At high $\mathrm{We}$, partial rebound is only identified on the most rigid surface at $\mathrm{We}\gtrsim 92$. Rebounding droplets behave similarly to elastic drops rebounding on superhydrophobic surfaces and the impact process is independent of surface viscoelasticity. Further, surface viscoelasticity does not influence drop spreading after impact—as the surfaces behave like rigid surfaces—but it does affect drop recoiling. Also, the postimpact drop oscillation on soft viscoelastic surfaces is influenced by dynamic wettability of these surfaces. Comparing sessile drop oscillation with a damped harmonic oscillator allows us to conclude that surface viscoelasticity affects the damping coefficient and liquid surface tension sets the spring constant of the system.