  We consider the problem of finding a 1-planar drawing for a general graph,
where a 1-planar drawing is a drawing in which each edge participates in at
most one crossing. Since this problem is known to be NP-hard we investigate the
parameterized complexity of the problem with respect to the vertex cover
number, tree-depth, and cyclomatic number. For these parameters we construct
fixed-parameter tractable algorithms. However, the problem remains NP-complete
for graphs of bounded bandwidth, pathwidth, or treewidth.
