The Museo Archeologico Regionale Eoliano, spread over several buildings inside the castle walls, is the best small archaeological museum in southern Italy and the single best reason to visit Lipari. Its strength is the prehistoric and Greek collections: obsidian tools from Neolithic Lipari (the obsidian trade made the island rich), Bronze Age pottery, an exceptional collection of Greek and Roman theatrical masks, and a complete reconstructed sea-trade-route gallery of shipwreck finds.
Allow at least two hours; longer if you read the labels. The labels are mostly in Italian, with patchy English. The site is hilly and involves stairs between buildings.