You can use Writer, Draw, or both to create many of the documents that you can create in Publisher. I've seen people recommend Scribus, but then list numerous disadvantages that make me skeptical of its suitability.
I would suggest using Writer's column, Frame, Section and Text Box features, along with its drawing tools, as a first approach for brochures and standard documents. Use Draw for items with more fixed dimensions, like postcards.
One of the most powerful features for switching from Publisher is linked frames. Writer allows you to link frames (Insert > Frame, then use the Frames toolbar that appears) to link one or more frames so that they flow together even when disconnected or on different pages.
This was first published in July 2006