A Smoothed Particle Hydrodynamics (SPH) solution to the Rayleigh-Taylor Instability(RTI) problem in an incompressible viscous two-phase immiscible fluid with surface tension is presented. To validate the numerical model for the RTI problem, simulation results are quantitatively compared with analytical solutions in linear regime. It is found that the SPH method slightly overestimates the border of instability. The long time evolution of simulations is presented for investigating changes in the topology of rising bubbles and falling spike in RTI. It is shown that the numerical algorithm used in this work is capable of capturing the interface evolution and growth rate in RTI accurately.