  This paper is devoted to the study of metric subregularity and strong
subregularity of any positive order $q$ for set-valued mappings in finite and
infinite dimensions. While these notions have been studied and applied earlier
for $q=1$ and---to a much lesser extent---for $q\in(0,1)$, no results are
available for the case $q>1$. We derive characterizations of these notions for
subgradient mappings, develop their sensitivity analysis under small
perturbations, and provide applications to the convergence rate of Newton-type
methods for solving generalized equations.
