Droplet impacts are common in many applications such as coating, spraying, or printing; understanding how droplets spread after impact is thus of utmost importance. Such impacts may occur with different velocities on a variety of substrates. The fluids may also be non-Newtonian and thus possess different rheological properties. How the different properties such as surface roughness and wettability, droplet viscosity, and rheology as well as interfacial properties affect the spreading dynamics of the droplets and the eventual drop size after impact are unresolved questions. Most recent work focuses on the maximum spreading diameter after impact and uses scaling laws to predict this. In this paper, we show that a proper rescaling of the spreading dynamics with the maximum radius attained by the drop and the impact velocity leads to a unique single and thus universal curve for the variation of diameter versus time. The validity of this universal functional shape is validated for different liquids with different rheological properties as well as substrates with different wettabilities. This universal function agrees with a recent model that proposes a closed set of differential equations for the spreading dynamics of droplets.