  Let $f:X\rightarrow Y$ be a map. $f$ is a {\it sequence-covering
map}\cite{Si1} if whenever $\{y_{n}\}$ is a convergent sequence in $Y$ there is
a convergent sequence $\{x_{n}\}$ in $X$ with each $x_{n}\in f^{-1}(y_{n})$;
$f$ is an {\it 1-sequence-covering map}\cite{Ls2} if for each $y\in Y$ there is
$x\in f^{-1}(y)$ such that whenever $\{y_{n}\}$ is a sequence converging to $y$
in $Y$ there is a sequence $\{x_{n}\}$ converging to $x$ in $X$ with each
$x_{n}\in f^{-1}(y_{n})$. In this paper, we mainly discuss the
sequence-covering maps on generalized metric spaces, and give an affirmative
answer for a question in \cite{LL1} and some related questions, which improve
some results in \cite{LL1, Ls4, YP}, respectively. Moreover, we also prove that
open and closed maps preserve strongly monotonically monolithity, and closed
sequence-covering maps preserve spaces with a $\sigma$-point-discrete
$k$-network. Some questions about sequence-covering maps on generalized metric
spaces are posed.
