  In this paper we define a new class of partially filled arrays, called
$\lambda$-fold relative Heffter arrays, that are a generalisation of the
Heffter arrays introduced by Archdeacon in 2015. After showing the connection
of this new concept with several other ones, such as signed magic arrays, graph
decompositions and relative difference families, we determine some necessary
conditions and we present existence results for infinite classes of these
arrays. In the last part of the paper we also show that these arrays give rise
to biembeddings of multigraphs into orientable surfaces and we provide infinite
families of such biembeddings. To conclude, we present a result concerning
pairs of $\lambda$-fold relative Heffter arrays and covering surfaces.
