Symmetry and topology dictate the properties of quantum systems, and their understanding has been a foundation for the classification and characterization of quantum materials. In this talk, I will introduce how the two interplay with each other and give rise to exotic phenomena such as localized defect modes that fundamentally alter the bulk material properties. I will introduce a new class of materials, denoted higher-order topological insulators (HOTIs), and show that they can host helical modes along screw or edge dislocations. When this occurs, the helical mode is necessarily bound to a dislocation characterized by a fractional Burgers vector, macroscopically detected by the existence of a stacking fault. The robustness of a helical mode on a partial defect is demonstrated by an adiabatic transformation that restores translation symmetry in the stacking fault. Since partial defects and stacking faults are commonplace in bulk crystals, the existence of such helical modes can measurably affect the expected conductivity and thermoelectric properties of these materials. Finally, I will describe a general framework towards the classification of symmetry breaking defects based on symmetry representations, and show how these defects can be instrumental to find other exotic phenomena in the solid-state, such as nonabelian Majorana fermions.