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Résumé : We consider the problem of drawing graphs with at most one crossing per edge. These drawings, and the graphs that can be drawn in this way, are called $1$-planar. Finding $1$-planar drawings is known to be ${\mathsf{NP}}$-hard, but we prove that it is fixed-parameter tractable with respect to the vertex cover number, tree-depth, and cyclomatic number. Special cases of these algorithms provide polynomial-time recognition algorithms for $1$-planar split graphs and $1$-planar cographs. However, recognizing $1$-planar graphs remains ${\mathsf{NP}}$-complete for graphs of bounded bandwidth, pathwidth, or treewidth.