This talk presents polynomial-degree-robust (p-robust) equilibrated a posteriori error estimates for H(curl), H(div) and H(divdiv) problems, based on H1 auxiliary space decomposition. The proposed framework employs auxiliary space preconditioning and regular/Helmholtz decompositions to decompose the finite element residual into H^{-1} residuals that are further controlled by classical p-robust equilibrated a posteriori error analysis. As a result, we obtain novel and simple p-robust a posteriori error estimates of H(curl)/H(div) conforming methods and mixed methods for the biharmonic equation. In addition, we shall present equilibrated residual error estimators for the biharmonic equation in primal form and elasticity equation in mixed form. The material is partially based on joint works with Ludmil Zikatanov and Han Shui.