  Global symmetries greatly enrich the landscape of topological quantum phases,
playing an essential role from topological insulators to fractional quantum
Hall effect. Topological phases in mixed quantum states, originating from
\textit{decoherence} in open quantum systems or \textit{disorders} in imperfect
crystalline solids, have recently garnered significant interest. Unlike pure
states, mixed quantum states can exhibit \textit{average symmetries} --
symmetries that keep the total ensemble invariant but not on each individual
state. In this work, we present a systematic classification and
characterization of average symmetry-protected topological (ASPT) phases
applicable to generic symmetry groups, encompassing both average and exact
symmetries, for bosonic and fermionic systems. Moreover, we formulate the
theory of average symmetry-enriched topological (ASET) orders in disordered
bosonic systems. Our systematic approach helps clarify nuanced issues in
previous literature and uncovers compelling new physics. Notably, we discover
that (1) the definition and classification of ASPT phases in decohered and
disordered systems exhibit subtle differences; (2) despite these differences,
ASPT phases in both settings can be classified and characterized under a
unified framework of defect decoration and spectral sequence; (3) this
systematic classification uncovers a plethora of ASPT phases that are
\textit{intrinsically mixed}, implying they can exclusively manifest in
decohered or disordered systems where part of the symmetry is average; (4)
similarly for ASET, we find intrinsically disordered phases exhibiting exotic
anyon behaviors -- the ground states of such phases necessarily contain
localized anyons, with gapless (yet still localized) excitation spectral.
