A new security patch is out for Magento 1, addressing 16 APPSEC issues: https://magento.com/security/patches/supee-9767
Seven of the vulnerabilities score 8.0 or higher for CVSSv3 Severity, and they are being exploited in the wild, so this is a critical patch. Sites can apply SUPEE-9767, or update to the new release CE 1.9.3.3 / EE 1.14.3.3.
These patches are rarely painless, so:
What are common issues or pitfalls to watch out for when applying SUPEE-9767?
UPDATE 2017-07-12:
Magento has released SUPEE-9767 V2 and CE 1.9.3.4 to address many of the issues from the initial patch. If you applied V1, you should revert and then apply V2. If you haven't patched yet, just apply V2, and most of the issues brought up here will not be relevant.