"No hanky-panky in the TARDIS." Well, sure, this is family television after all. But in sharp contrast to the old series, since 2005 the relationships between the characters have been just as important to the show as the Monster of the Week. This has introduced a new subculture to Doctor Who fandom: "Shipping."
(Of note: There's a fantastic interchange between new series producer Phil Collinson and Eighth Doctor companion Daphne Ashbrook on the subject toward the end of Doctor Who: Podshock 141 [live at Gallifrey 20].)