The Fibonacci sequence F(n) = (0,1,1,2,3,5,8,13,21,...), where F(0) = 0, F(1) = 1, and F(n) = F(n-1) + F(n-2) for n > 1, was discovered in 1202 and has been the object of much mathematical fascination for over 800 years. In this talk, we will search for Fibonacci numbers that have other interesting mathematical properties--perfect squares, triangular numbers, and the like. Several questions are completely solved, while others remain open even today. In addition, we will explore the interplay between experimental mathematics, as revealed by computer work, and the rigor necessary for a complete mathematical proof.