In the present work, we introduce a new version of Fibonacci and Lucas numbers which we will call non-Newtonian Fibonacci and non-Newtonian Lucas numbers. Also, we discuss a variety of some properties of them. Furthermore, we give some formulas and identities such as Binet’s formula, the d’Ocagne’s identity, Cassini’s identity and Gelin-Cesàro identity involving these new types and we find the generating functions for these numbers.