Going beyond the lowest order, the authors in [10, 11] and [12] have analysed second order RG flows in some detail by looking at gradient flow aspects, monotonicity etc. Earlier, Tseytlin [13], claimed to have proved the monotonicity of the σ -model RG flow to all orders in α ′ though some important objections (at the second order level) regarding monotonicity were pointed out in [11]. In our work here, we revisit 2nd order flows for various simple target manifolds and also analyse further higher order flows. We first treat the flow equations as geometric flows in their own right, and figure out the consequences thereof by looking for general solutions, solitons, singularities etc. Subsequently, we point out the behaviour of the flows in the perturbative, RG flow domain and investigate the effect of including the higher order terms. It is important to mention the classes of manifolds we consider in our discussion. These are essentially of two kinds–unwarped ones and warped product varieties. In the former case, we consider the simplest toy examples – namely the two sphere and two dimensional hyperbolic space. We look at the higher order flow equations and find exact solutions, from which we can arrive at useful conclusions. We also investigate an unwarped product briefly. Later, we look at warped product manifolds where the generic line element is assumed to be that of a bulk spacetime in a braneworld model with one extra dimension [14]. For all the manifolds under consideration gradient flow and monotonicity, upto second order in α ′ , holds without any problems [11]. Thus, our main focus in this work is two-fold: • Obtaining the perturbative corrections that arise due to the inclusion of higher order terms in the RG flow equations. • Studying the general geometric evolution, solitons, singularities etc. which arise when we treat the RG flow equations at various orders, as genuine geometric flows in their own right, without reference to the perturbative RG aspect. II. THE RG FLOW EQUATIONS AT VARIOUS ORDERS The RG flow equations for the background metric coupling g$_{ij}$ , in bosonic non-linear sigma -models are given by – ∂g$_{ij}$ ∂λ = − β$_{ij}$ (1) 3