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Author Series -- connected books from a single author. The Stephanie Plum series by Janet Evanovich is an example.
Multi-Author Series -- series where the books do connect in some way but are published by several authors. Star Wars is an example.
Publisher Series -- series in numerical order that are not necessarily related. Harlequin Presents is an example. Publisher imprints are also included in this category. Zebra Heartfire is an example.