Non-commutative algebraic geometry: why and how

Non-commutative algebraic geometry is a relatively new subject that has recently seen rapid progress. In particular, it turned out to be possible to prove some strong general theorems such as a non-commutative version of the Hodge-to-de Rham Degeneration. But the subject is still not too well known to the general mathematical audience. Avoiding technical details, I will try to give a general description of non-commutative algebraic geometry, with its motivations, applications, and results (both proved and conjectural).