In the talk, I will discuss the Turner-Turaev formalism for  unoriented Topological Quantum Field Theory (TQFT). Building upon this  formalism, I will introduce an analogous version of (d+1)-dimensional TQFT  for pro-p Poincaré duality groups.  In the case of d=1, this enables us to study cobordisms and TQFTs for both  the maximal pro-p quotient of absolute Galois groups of p-adic fields and  pro-p completions of fundamental groups of surfaces.  This generalisation gives a framework for arithmetic TQFTs and strengthens  the analogies within arithmetic topology, which relates p-adic fields to  surfaces (oriented mod p^r).  I will also provide an outline of the classification of such TQFTs for d=1,  in terms of Frobenius algebras with some extra structure.  The talk is based on joint work with Oren Ben-Bassat.