Shojo, the Japanese term for "young girls," is also used as a generic descriptor for anime aimed at that audience. The best such shows, though don't limit their audience appear to just girls or women: many of these titles have strong male fanbases, and the newer shojo shows cross over that much more readily to a broad viewership. Here's some of the best that reach out to male and female alike.
All titles are listed in alphabetical order.
The genteel Japanese poetry-guessing game of karuta -- still widely played in Japan as a competitive sport -- is used as the underpinning for a story about a girl, overshadowed by her bigger sister, who is determined to excel at something in her life. It's a fine example of how anime can be about "things Japanese" without being excessively insular; once we understand how the game works, it's all about the people.
4. Glass Mask
"Theatrical" is the perfect adjective for this story, an adaptation of a long-running classic shojo manga that's been unspooling since the 1970s. A young girl with ambitions to become an actress defies all odds -- including a monomaniacal mentor, jealous rivals, and the prejudices of acquaintances -- to fulfill her dreams. It's fundamentally a soap opera, but one assembled with such skill and conviction that it crosses over into full-blooded, unashamed Douglas Sirk territory.
Shy Sawako has never been the object of anyone's attention, let alone another boy -- until the day exactly that happens, and she realizes she hasn't the faintest idea how to deal with it. For support she turns to two other newly-minted female friends --both a whole order of magnitude more boisterous and vivacious than she is -- and begins the whole process of coming out of her shell without regrets. Derived from an equally well-observed manga of the same name, this show takes delight in all the nuances and subtleties of the emotions that run between people. So much so that even the "bad girl" of the bunch, who sets out to steal Sawako's would-be boyfriend from her, is a fully-realized character and not just a stock obstacle.
Three girls are wrenched out of our world and transported to another land where magic and adventure are the order of the day -- and where they discover their very presence there has potentially troubling implications. Should they stay, and use the force of their will to maintain the balance of the world, or endeavor to return home? Adapted from the manga by CLAMP (Cardcaptor Sakura, X, xxxHOLiC), it mixes swashbuckling with story elements from more conventional shojo (i.e., magical girl) stories and even mecha anime.
The Lillian Girls' Academy, a Catholic school in Tokyo, uses a system of apprenticeship where the older students mentor the younger ones. This arrangement proves more than a little flustering for newcomer Yumi, when she's taken under the wing of everyone's favorite "soeur", the tall and elegant Sachiko. One of the most popular examples of a yuri story (female-female relationships) -- so popular, in fact, it's been through four seasons.
Haruhi Fujioka has no idea what a mousy, anonymous girl like her is doing in the halls of the lush and prestigious Ouran Academy. To say nothing of what she's doing in the even lusher halls of the school's "Host Club", made up of a bevy of handsome young men whose sole job is to bring joy to the girls who drop in -- and who, like everyone else, assume Haruhi is a boy. And after she mistakenly breaks a valuable vase that she has no way in heck of ever paying for, they blackmail her into becoming one of them. What starts as side-splitting farce straight out of the screwball comedies of the Thirties, a version of the "reverse harem" story (one girl, many guys), develops some surprisingly emotional sides as well but never stops being funny, or for that matter relatable.
A foundational shojo anime from the 1960s, derived from Osamu Tezuka's equally foundational manga creation. A girl born with both male and female hearts takes up a sword to defend her kingdom when it becomes imperiled by dark forces. Sadly, the only extant version of the show for non-Japanese audiences is the edited and English-dubbed edition known as Choppy and the Princess.
Madoka Kaname is given the chance to become a magical girl and fight evil ... and discovers that the whole thing is nowhere nearly as easy as it seemed on her favorite TV show. The genius of this show is not just that it's a well-told revisitation of the magical girl concept; it's in how the characters themselves are aware of the concept as an in-universe presence, and how that affects their ability to live up to it (or not). Further bonus points awarded for the dazzling experimental-animation look of the underworld Madoka and her cohorts enter when they battle the bad guys.