How big an RSA key is considered secure today? I think 1024 bit RSA keys were considered secure ~5 years ago, but I assume that's not true anymore. Can 2048 or 4096 keys still be relied upon, or have we gained too much computing power in the meanwhile?

Edit: Lets assume an appropriate padding strategy. Also, I'm asking both about security of signatures and security of data encryption.
Since 2000, on a given $\text{year}$, no RSA key bigger than $(\text{year} - 2000) \cdot 32 + 512$ bits has been openly factored other than by exploitation of a flaw of the key generator (a pitfall observed in poorly implemented devices including Smart Cards). This linear estimate of academic factoring progress should not be used for choosing a key length so as to be safe from attacks with high confidence (or, equivalently, conforming to standards with that aim), a goal best served by this website on keylength.
The current factoring record is the 829-bit RSA-250 in late Feb. 2020, see the summary by the CADO-NFS team. That came shortly after the 795-bit RSA-240 in Dec. 2019, see the detailed paper.
I emphasize that the above is about attacks actually performed by academics. As far as we know, hackers have always been some years behind (see below). On the other hand, it is very conceivable that well-funded government agencies are many years ahead in the factoring game. They have the hardware and CPU time. And there are so many 1024-bit keys around that it is likely a worthwhile technique to be in a position to break these. It is one of the few credible and conjectured explanations for claims of a cryptanalytic breakthrough by the NSA. Also, dedicated hardware could change the picture someday; e.g. as outlined by Daniel Bernstein and Tanja Lange: Batch NFS (in proceedings of SAC 2014; also in Cryptology ePrint Archive, November 2014). Or in the distant future, quantum computers usable for cryptanalysis.
By 2020, the main practical threat to systems still using 1024-bit RSA to protect commercial assets often is not factorization of a public modulus; but rather, penetration of the IT infrastructure by other means, such as hacking, and trust in digital certificates issued to entities that should not be trusted. With 2048 bits or more we are safe from that factorization threat for perhaps two decades, with fair (but not absolute) confidence.
Factorization progress is best shown on a graph (to get at the raw data e.g. to make a better graph, edit this answer)


This also shows the linear approximation at the beginning of this answer, which actually is a conjecture at even odds for the [2000-2016] period that I made privately circa 2002 in the context of deciding if the European Digital Tachograph project should be postponed to upgrade its 1024-bit RSA crypto (still widely used today). I committed it publicly in 2004 (in French). Also pictured are the three single events that I know of hostile factorization of an RSA key (other than copycats of these events, or exploitation of flawed key generator):

The Blacknet PGP Key in 1995. Alec Muffett, Paul Leyland, Arjen Lenstra, and Jim Gillogly covertly factored a 384-bit RSA key that was used to PGP-encipher &quot;the BlackNet message&quot; spammed over many Usenet newsgroup. There was no monetary loss.

The French &quot;YesCard&quot; circa 1998. An individual factored the 321-bit key then used (even though it was clearly much too short) in issuer certificates for French debit/credit bank Smart Cards. By proxy of a lawyer, he contacted the card issuing authority, trying to monetize his work. In order to prove his point, he made a handful of counterfeit Smart Cards and actually used them in metro tickets vending machine(s). He was caught and got a 10 months suspended sentence (judgment in French). In 2000 the factorization of the same key was posted (in French) and soon after, counterfeit Smart Cards burgeoned. These worked with any PIN, hence the name YesCard (in French) (other account in English). For a while, they caused some monetary loss in vending machines.

The TI-83 Plus OS Signing Key in 2009. An individual factored the 512-bit key used to sign downloadable firmware in this calculator, easing installation of custom OS, thus making him a hero among enthusiasts of the machine. There was no direct monetary loss, but the manufacturer was apparently less than amused. Following that, many 512-bit keys (including those of other calculators) have been factored.


Note: 512-bit RSA was no longer providing substantial security by 2000-2005. Despite that, reportedly, certificates with this key size were issued until 2011 by official Certification Authorities and used to sign malware, possibly by means of a hostile factorization.