The invention relates to novel substituted bicyclo[3.1.0]hexanes of the formula ##STR1## in which R.sup.1 represents, e. g., hydroxyl or alkoxy,R.sup.2 to R.sup.8 represent, e. g., hydrogen or alkyl,R.sup.9 and R.sup.10 represent, e.g., methyl or methoxy,X represents, e.g., oxygen,Y represents, e.g., oxygen or sulphur andZ represents, e.g., CH or N,as well as salts of the free acid (R.sup.1 =OH) and their further functional derivatives, and furthermore to processes and novel intermediates for their preparation, and to their use as herbicides.In addition, a novel process is described for preparing certain 2-keto-bicyclo[3.1.0]hexanes, which are required as intermediates, starting from correspondingly substituted .alpha.,.beta.-unsaturated carbonyl compounds (of the methacrolein type) and .beta.-keto acid esters (of the acetoacetic acid ester type).