According to Badeker's Sicily (an old guide book), the small town of Bagheria is situated 12 km to the east of Palermo. In the 17th and 18th century, villas were built here by the aristocracy of Palermo who wanted to reside away from Palermo and didn't want to be too far away.
The most lavish is the Villa Valguarnera (just across the piazza from Villa Palagonia), we tried to see it but it wasn't opened to the public. Like many of the villas in the Bagheria and the palazzos in Palermo, most if not all of them are not opened to the public.
The Villa Palagonia was built in 1715 and Goethe visited it in 1787 and wrote about it. It was acquired by the Castronovo family (www.castronovo.org) in 1885 and remains in the family to this day. Thanks to the Castronovo heirs I was able to see Villa Palagonia.