  We present a brief survey of recent results on boundedness of some classical
operators within the frameworks of weighted spaces $L^{p(\cdot)}(\varrho)$ with
variable exponent $p(x)$, mainly in the Euclidean setting and dwell on a new
result of the boundedness of the Hardy-Littlewood maximal operator in the space
$L^{p(\cdot)}(X,\varrho)$ over a metric measure space $X$ satisfying the
doubling condition. In the case where $X$ is bounded, the weight function
satisfies a certain version of a general Muckenhoupt-type condition For a
bounded or unbounded $X$ we also consider a class of weights of the form
$\varrho(x)=[1+d(x_0,x)]^{\bt_\infty}\prod_{k=1}^m w_k(d(x,x_k))$, $x_k\in X$,
where the functions $w_k(r)$ have finite upper and lower indices $m(w_k)$ and
$M(w_k)$.
  Some of the results are new even in the case of constant $p$.
