Such a leak could potentially expose stored passwords and other sensitive data, including personal photos, emails and instant messages.
Intel has highlighted its quick response to the exploits, which it learned about a year ago but only acknowledged on Wednesday after the existence of the Meltdown flaw began to leak through news websites.
Despite concern that fixes may slow down devices, Apple said its steps to address the Meltdown issue haven't dented performance.
Here's a look at what's affected, what's being done about it and whether you should worry.
The problems are based on recently discovered vulnerabilities with chips built into most of the world's computers, phones and work PCs.
The other flaw, called Meltdown, affects most Intel processors made after 1995.
Another variant of Spectre, the company said, has "a near zero risk of exploitation" on its processors.
Although companies continue to roll out fixes, experts say hackers may soon attempt to exploit the flaws on devices that haven't been updated.
Most of the immediate fixes will be limited to the Meltdown bug.
"In the coming days we plan to release mitigations in Safari to help defend against Spectre", said Apple on Friday.
For those technically minded, you can review the details of the security flaw that effects nearly all modern processors, over at Google's Project Zero blog. The ARM design is also used in Apple's mobile chips.
Apple has released three new security updates aimed at protecting Safari and WebKit from the Spectre attack.
Amazon said Wednesday in a blog post that "all but a small single-digit percentage of instances" of its EC2 systems, a service under its cloud computing platform, had already been protected, and urged customers to patch their operating systems using available updates.
Unauthorized access will be hard to detect so cloud-computing providers need to act quickly to protect against these vulnerabilities, said Ryan Kalember, senior vice president of cybersecurity at Proofpoint.
There are limits to what consumers can do now to protect their computers.
Consumers can mitigate the underlying vulnerability by making sure they patch up their operating systems with the latest software upgrades.
"All future Arm Cortex processors will be resilient to this style of attack or allow mitigation through kernel patches", the company said in a statement. 2 supplemental update for all supported devices. This isn't something you need to debate about, get the update.