We analyze Ricci flow (normalized/unnormalized) of product manifolds - unwarped as well as warped, through a study of generic examples. First, we investigate such flows for the unwarped scenario with manifolds of the type double-struck S signn × double-struck S signm, double-struck S signn × ℍm, ℍm × ℍn and also, similar multiple products. We are able to single out generic features such as singularity formation, isotropization at particular values of the flow parameter and evolution characteristics. Subsequently, motivated by warped braneworlds and extra dimensions, we look at Ricci flows of warped spacetimes. Here, we are able to find analytic solutions for a special case by variable separation. For others, we numerically solve the equations and draw certain useful inferences about the evolution of the warp factor, the scalar curvature as well as the occurrence of singularities at finite values of the flow parameter. We also investigate the dependence of the singularities of the flow on the initial conditions. We expect our results to be useful in any physical/mathematical context where such product manifolds may arise. © 2010 World Scientific Publishing Company.