  Let $\mathcal{M}$ be a von Neumann algebra equipped with a faithful normal
semi-finite trace $\tau$ and let $S_0(\tau)$ be the algebra of all
$\tau$-compact operators affiliated with $\mathcal{M}$. Let $E(\tau)\subseteq
S_0(\tau)$ be a symmetric operator space (on $\mathcal{M}$) and let
$\mathcal{E}$ be a symmetrically-normed Banach ideal of $\tau$-compact
operators in $\mathcal{M}$. We study (i) derivations $\delta$ on $\mathcal{M}$
with the range in $E(\tau)$ and (ii) derivations on the Banach algebra
$\mathcal{E}$. In the first case our main results assert that such derivations
are continuous (with respect to the norm topologies) and also inner (under some
mild assumptions on $E(\tau)$). In the second case we show that any such
derivation is necessarily inner when $\mathcal{M}$ is a type $I$ factor. As an
interesting application of our results for the case (i) we deduce that any
derivation from $\mathcal{M}$ into an $L_p$-space, $L_p(\mathcal{M},\tau)$,
($1<p<\infty$) associated with $\mathcal{M}$ is inner.
