Crows are little black scavengers. They have wings but no halo, therefore they are birds. (Actually, buildings can have wings too but let's not stray from the topic for now.) And quite intelligent for that matter (see ). Murphy, OHBB member (and maintainer of this wiki) claims to be one of them.
Appearances in the series
Crows play an integral part in the symbolism of Haibane Renmei, mainly from episodes 1 through 8.
episode 1: In Rakka's dream of falling from the sky, a crow flies to her rescue, trying to stop her by holding onto her gown. Rakka points out the futility of such an attempt and thankfully dismisses it (the crow). The aforementioned crow, probably deeply disturbed by such fatalistic attitudes, flees accordingly.
episode 2: After Kuu's little incident at the Great Gate Square, while the Old Home haibane are leaving, a crow calls out, or so it seems, to Rakka. Washi sends off the crow, which follows the Haibane for a while before flying off. We are also given a hint, on the first occasion, of Kana's apparent hatred towards them.
episode 3: While Rakka and Kuu are in town, they see a shop owner chasing them away. The crows, not the haibane. The latter would probably add some comic relief value but I digress. Kuu comments on the townspeople's opposition to crows and wishes they could talk with them. Rakka agrees.
episode 4: A slightly surreal crow with glowing cyan eyes appears in Rakka's dream at the beginning of the episode (00:01:33). Later, Kana chases two crows that are scavenging the contents of the trash incinerator at Old Home. While Kana and Rakka are standing on the balcony at the top of the clock tower, they see birds flying over the walls. Kana comments on this, mentioning the legend that crows are believed to carry lost items left behind by the haibane before they went into the cocoons. At the end of the episode (00:21:21) a crow poops on Kana.
episode 5: Amazingly enough, no crows actually appear in this episode (although they are heard at times).
episode 6: They appear several times during the storm. When Rakka looks out of the window (00:13:06) a crow descends onto the balcony's railing, caws at her in a presumably friendly manner then flies away. Not much later (00:13:33) Rakka leans out of another window in an attempt to trace its path, which apparently leads to the Western Woods. When Kuu initiates her Flight, a whole bunch of crows flee from the Woods.
episode 7: Upon Rakka's arrival at Old Home (00:09:27), two crows, sitting on a tree, call out, presumably at her, then fly away, leaving fallen feathers behind.
episode 8: When Reki and Rakka take out the Young Feathers for a walk to the thrift shop, Rakka notices two crows sitting on the power lines. (This happens shortly after she mentions not completely remembering what happened in her cocoon dream.) Later (at 00:13:27), when she is crying on the Hill of Winds, a lone crow, sitting atop a windmill, calls out to her. This finally prompts Rakka to enter the Western Woods. At 00:15:57 she finds a bunch of them gathered at a disused well deep in the woods. Down in the well she finds the skeleton of a crow, and she feels that this is the one that called her several times.
Interpretation in the series' context
As said by Kana (and later, Washi), crows are symbolic of what the haibane left behind in their former lives. In Rakka's case, this is quite literal, as the crow she meets in her cocoon dream, and later, in the well, is someone from her previous life who cared for her, though she failed to notice that. It is possible that this person died as well while trying to rescue Rakka, although this is only implied. Several theories were proposed on the crow's true identity - see Threads. When Kuu took her Flight, many crows were gathered in the Western Woods. This might imply that they return the forgotten memories of haibane who go over the Wall.
In mythology crows tend to represent the spiritual aspect of death and the transition of spirit to the afterlife. More to come later.
Outside the series
Crows appear in the opening sequences of two other series Yoshitoshi ABe was involved in: Serial Experiments Lain and, briefly, in the opening of NieA_7. Also see Creators. They are considered by some to be a "signature" of his, akin to, for example, the basset hound in the works of Mamoru Oshii.
In fanfiction FHD Remix, the crows are identified as another possibility for reincarnated/resurrected beings (the other one being the haibane.) Such crows retain a complete memory of their past lives, and thus their original names, like Nancy Vaughn and Jack Campbell. Their behaviour in Haibane Renmei is consistent with this conjecture.