The Drosophila connectome, once completed, will be a complete list of the roughly 135,000 neurons in the brain of the fruit fly Drosophila melanogaster, along with all of the connections (synapses) between these neurons. As of 2013, the Drosophila connectome is a work in progress, being obtained by the methods of neural circuit reconstruction. Portions of two of the 76 compartments of the Drosophila brain have connectomes available, and others are subjects of ongoing study.
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Why Drosophila
Connectome research (connectomics) has a number of competing objectives. On the one hand, investigators prefer an organism small enough that the connectome can be obtained in a reasonable amount of time. This argues for a small creature. On the other hand, one of the main uses of a connectome is to relate structure and behavior, so an animal with a large behavioral repertoire is desirable. It's also very helpful to use an animal with a large existing community of experimentalists, and many available genetic tools. Drosophila looks very good on these counts:
Current status
A high-level connectome, at the level of brain compartments and interconnecting tracts of neurons, exists for the full fly brain. A version of this is available online.
Detailed circuit-level connectomes exist for the lamina and a medulla column, both in the visual system of the fruit fly.