When air is rapidly injected into a thin layer of oil contained between two closely spaced plates, the resultant air bubble, instead of growing as a circle of uniform radius, develops fingers that penetrate into the oil layer. The finger tips repeatedly split, resulting in a complex multifingered structure. We investigate the geometrical properties of these structures, which are fractals (they have non-integer dimension). The fractal structures observed in our laboratory experiments are found to exhibit the same geometrical and dynamical properties as a model system called Diffusion Limited Aggregation. Growing bacterial colonies and flame fronts can also produce similar fractal structures. We also examine the wavy edges of torn plastic garbage bags and leaves. We find that the edges of these thin sheets can be fractal, and the fractal properties can be related to the metrics of the torn garbage bags and leaves.