  The paper is devoted to so-called local and 2-local derivations on the
noncommutative Arens algebra $L^\omega(M, \tau)$ associated with a von Neumann
algebra $M$ and a faithful normal semi-finite trace $\tau.$ We prove that every
2-local derivation on $L^\omega(M, \tau)$ is a spatial derivation, and if $M$
is a finite von Neumann algebra, then each local derivation on $L^\omega(M,
\tau)$ is also a spatial derivation and every 2-local derivation on $M$ is in
fact an inner derivation.
