A left-truncatable prime stays prime each time you remove its leading digit: 9137 → 137 → 37 → 7. No digit may be 0 (a truncation would then have a leading zero). There are exactly 4260 of them in base 10, the largest being the 24-digit 357,686,312,646,216,567,629,137.
They're the left-handed mirror of the [[right-truncatable-prime]]s, and the two families overlap in the elegant set of two-sided truncatable primes, of which there are just fifteen (the largest being 739,397).