have you always wondered how the banach-tarski paradox - namely cutting a ball into pieces and assembling those pieces into two balls - actually works? or what the mathematical reason behind it is? in this talk we will answer those questions by 1) reformulating the banach-tarski paradox, 2) explaining what paradoxical decompositions of various objects are, 3) constructing one for a ball, and 4) motivated by all those things introducing the concept of amenability, which has its origin in the aforementioned paradox and is ubiquitous in many areas of mathematics nowadays.