A Sierpiński number is an odd integer k with the property that k · 2ⁿ + 1 is composite for all positive integer values of n. A Riesel number is defined similarly; the only difference is that k · 2ⁿ - 1 is composite for all positive integer values of n. In this paper we find Sierpiński and Riesel numbers among the terms of the well-known Fibonacci sequence. These numbers are smaller than all previously constructed examples. We also find a 23-digit number which is simultaneously a Sierpiński and a Riesel number. This improves on the current record established by Filaseta, Finch and Kozek in 2008. Finally, we prove that there are infinitely many values of n such that the Fibonacci numbers F_n and F_n+1 are both Sierpiński numbers.