The 320xx was the predecessor of the Swordfish microprocessor.
The first chip in the series was originally called 16032, later renamed 32016. It became available in the late 1970s, and may have been the first 32 bit chip to reach mass production and sale (at least according to National Semiconductor marketing hype.)
This chip had a 16 bit external databus, 24 bit external address bus, and a full 32 bit instruction set. The instruction set was extremely complex but mostly regular, with a large set of addressing modes. It was somewhat similar in spirit to (but not compatible with) the popular DEC VAX minicomputer instructions set.
National Semiconductor also produced related chips for floating point, memory management, and DMA. With the full set plus memory and peripherals, it was feasible to build a 32 bit computer system capable of supporting modern multi-tasking operating systems, something that had previously been possible only on expensive minicomputers and mainframes.
The 32032 arrived soon afterwards. It was almost completely compatible, but featured a 32 bit data bus (and i think a 31 bit address bus?) for somewhat faster performance. Both of these early chips were notorious for being unrealiable. There was a somewhat better chance of getting them to work if a full set of CPU, MMU, FPU, DMA chip were purchased as a matched, tested set, from Natsemi. Nonetheless, reliability trouble made the early 320XX fairly unpopular, and Natsemi were forced to sell them at much lower prices than the competing Motorola 68000 in order to sell any at all. This low price did at least make them somewhat popular with hobbiests wanting to build 32 bit computers on a very small budget.
During the 1980s, successor chips called the 32332 and 32532 arrived, maintaining a good degree of compatibility, with much improved reliability and performance. By then the damage to reputation had been done, and these chips were (probably unjustly) ignored by most of the market.
The Swordfish was a further advance, aimed at embedded systems and arriving circa 1990. They had some success in the laser printer market, despite intense competition from AMD and Intel RISC chips.