Primality testing was essentially solved in the 70s with Miller-Rabin. AKS made that (randomized) algorithm deterministic, albeit at much higher (polynomial) running-time.

For your overall question, the current record-holders for integer factorization wrote a paper on this a few years ago that is probably a good reference

https://hal.science/hal-03691141/file/cryptography.pdf

The (rough) outline of the paper is that

1. theoretically there's been no progress on factoring in ~30 years

2. practically, there have been both improved hardware + efficient implementations driving the progress. They estimate that current nation-states can (classically) break RSA-1024. The cost would be approximately 500,000 core-years of computation. At current cloud prices this is doable on aws for < $1B.

3. attacks against factoring use a technique ("index calculus") that can also be used to attack finite-field discrete logarithm. There were significant advances on that problem in the 2010s (at least for certain parameters, namely the "small characteristic" setting). An easy way to communicate this is that the RSA factoring record is ~830 bits, while the binary-field discrete logarithm record is > 30,000 bits. These significant advances have not been able to be ported over to factoring, nor have they been ported over to medium/large-characteristic discrete logarithm. It is a (very upsettingly) large open question of whether similar-magnitude improvements are possible more generally for index calculus algorithms.