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  1. We initiate the study of Simultaneous Graph Embedding with Fixed Edges in the beyond planarity framework. In the QSEFE problem, we allow edge crossings, as long as each graph individually is drawn quasiplanar, that is, no three edges pairwise cross. %We call this problem the QSEFE problem. We show that a triple consisting of two planar graphs and a tree admit a QSEFE. This result also implies that a pair consisting of a 1-planar graph and a planar graph admits a QSEFE. We show several other positive results for triples of planar graphs, in which certain structural properties for their common subgraphs are fulfilled. For the case in which simplicity is also required, we give a triple consisting of two quasiplanar graphs and a star that does not admit a QSEFE. Moreover, in contrast to the planar SEFE problem, we show that it is not always possible to obtain a QSEFE for two matchings if the quasiplanar drawing of one matching is fixed.