Abstract:  Ground states of gapped Hamiltonians can form 'symmetry-protected topological phases', characterized by zero-energy edge modes. We explore the quantum critical points between such topological phases in one spatial dimension. Two main questions are addressed, namely how universal properties of the critical point are related to the nearby gapped phases, and whether critical points themselves can be topologically non-trivial. The first question leads to a topological lower bound on the central charge. The second question is answered in full for a class of non-interacting fermions (BDI), with the critical phases being classified by a topological invariant. Curiously, this invariant protects exponentially localized edge modes, even though the bulk contains no massive degrees of freedom.