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We will give a survey of the last 15 years of research done in a particular example of non-commutative rings flourishing from the fact that free modules over some non-commutative rings can have two bases with different cardinality. Surprisingly enough not only non-commutative ring theorists, but also C*- algebraists gather together to advance the work done. The interplay between the topics stimulates interest and many proof techniques and tools are used from symbolic dynamics, ergodic theory, homology, K-theory and functional analysis. Open problem pages are put up and research schools are organized throughout the world. Over 150 papers have been published on this structure, so called Leavitt path algebras, which is constructed on a directed graph.