The spreading of drops on surfaces is ubiquitous and has relevance to many technological applications. In this work, we present two-dimensional numerical simulations of the surface tension driven spreading of drops dispensed on a fluid-fluid interface. A comprehensive picture describing the equilibrium shapes of the drops is provided in the form of a state diagram. We show that the analysis of kinetics of drops that spread symmetrically on the fluid-fluid interface reveal several interesting features: (i) the existence of a single length scale that describes the spreading process, (ii) the power law dependence of the temporal variation of the geometrical parameters of the spreading drop, (iii) the linear dependence of the power law exponents on the equilibrium enclosing angle of the liquid drop, (iv) a strong dependence of the power law exponents on the spreading coefficient, and (v) a collapse of the spreading kinetics data into a master curve. Though restricted to two dimensions, our analysis provides a rationale for explaining experimentally determined power law exponents which have been reported to vary over a wide range and hence to understand the universal nature of the spreading process. © 2020 American Physical Society.