Quantum states may exhibit asymmetry with respect to the action of a given
group. Such an asymmetry of states can be considered as a resource in
applications such as quantum metrology, and it is a concept that encompasses
quantum coherence as a special case. We introduce explicitly and study the
robustness of asymmetry, a quantifier of asymmetry of states that we prove to
have many attractive properties, including efficient numerical computability
via semidefinite programming, and an operational interpretation in a channel
discrimination context. We also introduce the notion of asymmetry witnesses,
whose measurement in a laboratory detects the presence of asymmetry. We prove
that properly constrained asymmetry witnesses provide lower bounds to the
robustness of asymmetry, which is shown to be a directly measurable quantity
itself. We then focus our attention on coherence witnesses and the robustness
of coherence, for which we prove a number of additional results; these include
an analysis of its specific relevance in phase discrimination and quantum
metrology, an analytical calculation of its value for a relevant class of
quantum states, and tight bounds that relate it to another previously defined
coherence monotone.