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How can this be done? It looks like the invitations are raised. Is it with drop shadow or another tool? I'd like to be able to do this in Illustrator.
Illustrator has a (raster) drop shadow effect:
Choose Effect > Stylize > Drop Shadow
Effect > Stylize > Drop Shadow
You'll also want to ensure the Document Raster Effects Setting is correct:
Document Raster Effects Setting
This is a raster effect and will create embedded raster images upon output or flattening.
Required, but never shown