It is proved that the algebra of regular functions on quantum $m\times n$ matrices admits a division ring of quotients and that this division ring is a division ring of twisted rational functions. A description is given of the field of central elements in the division ring of rational functions on quantum $m\times n$ matrices in the one-parameter and multiparameter cases. In the one-parameter case for $q$ of a general form the center is a purely transcendental extension of a field $\mathbb K$ of degree $l$ (were $l$ is the greatest common divisor of $m$ and $n$) if both numbers $m/l$ and $n/l$ are odd. If at least one of the numbers $m/l$ and $n/l$ is even, then the center is scalar. In the multiparameter case the answer depends upon the parameters $P$,$Q$, $c$. Here the generators of the center are described and it is proved that the center is scalar for the case of even $n$ and parameters of a general form. Analogous result are obtained for the division ring of rational functions on a quantum Borel subgroup of $GL_{P,Q,c}(n)$.