The shift towards cheaper, non-platinum group metal electrocatalyst materials for clean energy technologies is coupled with challenges in maintaining long-term performance. For practical purposes, electrocatalytic stability typically focuses on catalyst electrochemical performance over time. However, a deeper understanding of catalyst material property changes during operation is needed to enable material-specific design strategies for long-term stabilization. In the last several decades, improvements in material characterization techniques have made it possible to probe the composition, structure, and degradation products of catalysts in situ/operando. Herein we review the current understanding of in situ/operando material stability of Co-based electrocatalysts for the oxygen evolution (OER) and reduction reactions (ORR) in acidic and alkaline environments. We focus on in situ/operando materials characterization of three categories of Co-based OER/ORR catalysts: oxides and (oxy)hydroxides, mixed-metal catalysts, and non-oxide materials. This review aims to compile and compare the results from multiple studies and techniques to provide insight into the role that the starting material, pH, and applied potential have on the active surface and stability of Co-based materials. We conclude by highlighting directions that have been underexplored with opportunities for continued research including improving in situ/operando characterization, methods to probe long-term material changes, and the development of operando studies on full-scale devices. Using Co-based materials as a case study, this review shows the vital role that in situ/operando characterization must play in the future of improving electrocatalyst stability.