Thispaperdealswithabsorbingboundaryconditionsforhyperbolicsystemsinoneandtwospacedimensions.Weprovethestrictwell-posednessoftheresultinginitialboundaryvalueproblemin1D.AfterwardsweestablishtheGKS-stabilityofthecorrespondingLax-Wendroff-typefinitedifferencescheme.Hereby,wehavetoextendtheclassicalproofs,sincethe(discretized)absorbingboundaryconditionsdonotfitthestandardformofboundaryconditionsforhyperbolicsystems.