Abstract: Power laws! Fractals! Networks! They're everywhere!
So, why should we care? More than fashionable, the implications of distributions that do not have a well-defined mean (and thus, a characteristic scale) are deep and have yet to be fully acknowledged. For example, calculus requires a well defined mean, so it should not be used for phenomena with scale-free-ish distributions.
I will review examples and tools that have been used to describe phenomena that are "free of scale" or "slaves of scale". Even when they are fundamentally different, both classes can be described in terms of structural balance.