Mesopotamian religion is the oldest that can be read in its own words, and the reason is an accident of materials. Cuneiform was impressed into clay, and clay that has been through a fire becomes ceramic and survives indefinitely. Libraries burned, and burning is what preserved them.
The single richest recovery is the collection assembled at Nineveh under Ashurbanipal in the seventh century BC, excavated in the mid nineteenth century and now largely in the British Museum. It supplied the flood narrative that reached the public in 1872 and immediately changed the terms of European discussion about Genesis. Temple archives, omen compendia, hymns, incantations and the astronomical diaries fill out the picture from many other sites.
The bias in the record is scribal and institutional. These are the documents of temple and palace bureaucracies, produced by a small trained class, so ritual procedure and state theology are well represented and private devotion much less so. Letters and personal names are among the few windows onto what an ordinary person thought the gods were for.
Two practical constraints remain. Excavation has recovered far more tablets than have been published, so a substantial part of the corpus has never been read by anyone. And decipherment was itself contested in the nineteenth century and settled only by controlled tests, which is why the earliest translations should be used with care.