In Nature
There are many examples of Fibonacci numbers (numbers that appear in the sequence) appearing in the natural world. However, just because a series of numbers can be applied to an object, that doesn't imply there's a correlation between the math and reality.
Fibonacci numbers do appear in nature often enough to prove they reflect some naturally occurring patterns. You can commonly spot these by studying the manner in which various plants grow.
Many seed heads, pinecones, fruits and vegetables display spiral patterns that when counted express Fibonacci numbers. Look at spirals of seeds in the center of a sunflower and you'll observe patterns curving left and right. If you count these spirals, your total will be a Fibonacci number. Divide the spirals into those pointed left and right and you'll get two consecutive Fibonacci numbers. You can decipher spiral patterns in pinecones, pineapples and cauliflower that also reflect the Fibonacci sequence in this manner.