The reports are vary, some users report they face delay in sending text messages, some others are totally unable to send any messages and receive error message saying "not delivered".
Things getting worse because Apple doesn't seem to detect that there is a problem on their text messaging app, by saying no issue with iMesssage.
How to fix the iMessage problem: "not delievered".
You can try this quick fix, which is disable and enable the iMessage app. To do so, you can head to Settings app, then look down to find the Messages, tap that, then tap on iMessage toggle to disable, wait for a while then enable it again.
This is a temporary solution, until Apple comes up with the fix. And hope this helps you out from the iMessage error.