It started life as the Application System/400 ... or AS/400.
It combined the best of S/38 and S/36, leaning on S/38 for OS foundation and S/36 for user-friendliness. S/36 and S/38 in turn, have a rich heritage in the S/34 which in its hey day had a revolutionary 48 bit OS at a time when 8 bit was standard for PCs.
Many IBM customers on S/36 SSP were slow to embrace the AS/400 so IBM introduced the Advanced System 36 ... or AS/36 which had two models, the AS/236 and AS/436. The AS/436 could run SSP or OS/400 with SSP as a guest OS and had an upgrade path to AS/400 models. Interestingly the AS/236 was the first 64 bit and RISC based entry in the AS/400 family.
In 2006 the name changed again. This time the family name changed to System i which incorporates all systems from AS/400 throuh AS/400e to iSeries and including today's hardware. The current machine in the family is the the System i5. Theoretically the next generation will be the System i6 when the POWER6 chips are released.