We discuss some recent and ongoing works on the dynamics of flows with various expansive measures. In particular, we present a measurable version of the Smale’s spectral decomposition theorem for flows. More precisely, we prove that if a flow φ on a compact metric space X is invariantly measure expanding on its chain recurrent setCR(φ) and has the invariantly measure shadowing property on CR(φ) then φ has the spectral decomposition, i.e. the nonwandering set Ω(φ) is decomposed by a disjoint union of finitely many invariant and closed sets on which φ is topologically transitive. Moreover we show that if φ is invariantly measure expanding on CR(φ) then it is invariantly measure expanding on X. Using this, we characterize the measure expanding flows on a compact C∞ manifold via the notion of Ω-stability. This is joint work with N. Nguyen.