  This paper deals with a parabolic-elliptic chemotaxis-consumption system with
tensor-valued sensitivity $S(x,n,c)$ under no-flux boundary conditions for $n$
and Robin-type boundary conditions for $c$. The global existence of bounded
classical solutions is established in dimension two under general assumptions
on tensor-valued sensitivity $S$. One of main steps is to show that $\nabla
c(\cdot,t)$ becomes tiny in $L^{2}(B_{r}(x)\cap \Omega)$ for every
$x\in\overline{\Omega}$ and $t$ when $r$ is sufficiently small, which seems to
be of independent interest. On the other hand, in the case of scalar-valued
sensitivity $S=\chi(x,n,c)\mathbb{I}$, there exists a bounded classical
solution globally in time for two and higher dimensions provided the domain is
a ball with radius $R$ and all given data are radial. The result of the radial
case covers scalar-valued sensitivity $\chi$ that can be singular at $c=0$.
