Summary: We study fundamental properties and applications of the, so called, $\delta$-Fibonacci numbers $a_n(\delta)$ and $b_n(\delta)$. For these numbers, many special identities and interesting relations can be generated. Also the formulas connecting the numbers $a_n(\delta)$) and $b_n(\delta)$ with Fibonacci and Lucas numbers are presented. Moreover, some polynomials generated by $a_n(\delta)$) and $b_n(\delta)$ are discussed.