We will briefly review  Wilson-Kadanoff type renormalization group (RG) maps for Ising spin systems (often called real-space RG maps) and the lack of progress in proving that there is a non-trivial fixed point for these maps. The Ising model can be written as a tensor network, and RG maps can be defined in the tensor network formalism. In joint work with Slava Rychkov we proved that in two dimensions for a particular tensor network RG map the high temperature fixed point is locally stable. We are currently working on using such RG maps for tensor networks to study the low temperature phase. Here there are two stable fixed points corresponding to the two ground states and one unstable fixed point which is related to behavior near the phase coexistence curve. We hope that these results are a modest start towards proving the existence of a non-trivial fixed point for a tensor network RG map which would correspond to the critical point of the Ising model.