1. Literally, to grasp something, such as a cliff, with one's fingertips to avoid falling. The stranded hiker was hanging on by her fingertips until the rescue crew arrived.

2. By extension, to narrowly avoid problems or failure. They're hanging on by their fingertips out there—the other team's offensive is totally overwhelming them. Now that I have three small children to care for, I feel as if I'm hanging on by my fingertips every day.