  Lomonaco and Kauffman developed knot mosaics to give a definition of a
quantum knot system. This definition is intended to represent an actual
physical quantum system. A knot $n$-mosaic is an $n \times n$ matrix of 11
kinds of specific mosaic tiles representing a knot or a link. The mosaic number
$m(K)$ of a knot $K$ is the smallest integer $n$ for which $K$ is representable
as a knot $n$-mosaic. In this paper we establish an upper bound on the mosaic
number of a knot or a link $K$ in terms of the crossing number $c(K)$. Let $K$
be a nontrivial knot or a non-split link except the Hopf link. Then $m(K) \leq
c(K) + 1$. Moreover if $K$ is prime and non-alternating except $6^3_3$ link,
then $m(K) \leq c(K) - 1$.
