Beings from the spirit world are everpresent in Anime (and Manga). Miyazaki’s movies masterpieces (I haven’t seen a bad one to date) like Princess Mononoke and Spirited Away strongly feature spirits.
In Princess Mononoke, it’s basically all about spirits: the movie begins with an attack on a village let by the boar God Nago. Ashitaka is eventually cursed by Nago and as he journeys to find a cure for his curse, he mets San and all the spirits of the forest.
Spirited Away tells the story of Chihiro (renamed Sen by Yubaba) as she has to work in a bathhouse for … spirits. There she meets up with Haku (the spirit of river Kohaku) whom she frees him from Yubaba’s grasp and ‘No face’.
The same can be said of the popular series that are being (or have been) broadcasted. Bleach, for example, is all about spirit – Ichigo (the main character) is permanently in the spiritual world and even becomes a “part-time” shinigami (Death God) – his job as a shinigami is to send souls to the Soul Society. Yet, all anime do not feature spirits the same way. The shinagamis of Death Note are quite peculiar: they look more like a bunch of punks keen on gambling 🙂 have as job to decide when and how someone should die. Though spirits (or characters) from different anime tend to look alike, they are quite different – a Gillian type Menos in Bleach is certainly no ‘No face’.
All this (to me) seems like the Kami that we have in the Shinto religion – except for the shinigami stuff which I’ve heard has been imported from Europe. Am I right? Well I’m not “into” Japan that much to answer but the Japanese Culture and its influence on anime does fascinate me.
Random thoughts ... 
bleach rulezzzzz
I am not a “connaisseur” but I do feel some kind of mysticism in all those Miyazaki masterpieces… It’s more profound than religion (which is man-made IMHO).