  We investigate classification and detection of entanglement of multipartite
quantum states in a very general setting, and obtain efficient $k$-separability
criteria for mixed multipartite states in arbitrary dimensional quantum
systems. These criteria can be used to distinguish $n-1$ different classes of
multipartite inseparable states and can detect many important multipartite
entangled states such as GHZ states, W states, anti W states, and mixtures
thereof. They detect $k$-nonseparable $n$-partite quantum states which have
previously not been identified. Here $k=2,3,\cdots,n$. No optimization or
eigenvalue evaluation is needed, and our criteria can be evaluated by simple
computations involving components of the density matrix. Most importantly, they
can be implemented in today's experiments by using at most $\mathcal{O}(n^2)$
local measurements.
