  Mosaic diagrams for knots were first introduced in 2008 by Lomanoco and
Kauffman for the purpose of building a quantum knot system. Since then, many
others have explored the structure of these knot mosaic diagrams, as they are
interesting objects of study in their own right. Knot mosaics have been
generalized by Gardu\~{n}o to virtual knots, by including an additional tile
type to represent virtual crossings. There is another interpretation of virtual
knots, however, as knot diagrams on surfaces, which inspires this work. By
viewing classical mosaic diagrams as $4n$-gons and gluing edges of these
polygons, we obtain knots on surfaces that can be viewed as virtual knots.
These virtual mosaics are our present objects of study. In this paper, we
provide a set of moves that can be performed on virtual mosaics that preserve
knot and link type, we show that any virtual knot or link can be represented as
a virtual mosaic, and we provide several computational results related to
virtual mosaic numbers for small classical and virtual knots.
