The IAA understands the “Assyriology” in its name to refer to the study of ancient Mesopotamia and neighbouring regions through textual, archaeological and art historical approaches. Scholars usually specialise in either textual or archaeological studies, and often find themselves in different departments. Nevertheless, specialists often work closely together, especially on excavations.
A more detailed description of textual study (Assyriology in its narrower sense) can be found here. A more detailed description of the archaeological study can be found here.
There are many useful recordings of specialists freely available online.
If you would like to go and see objects from ancient Mesopotamia, a list of museums with relevant displays can be found here.