On main conjectures in non-commutative Iwasawa Theory

Recently the main conjecture of non-commutative Iwasawa theory for totally 
real fields was proven under the assumption of vanishing of certain
$\mu$ invariant. The proof reduces the non-commutative main conjecture
to a family commutative main conjecture (which are known due to Wiles) and 
certain congruences between special values of Artin $L$-functions (which
are proven using the Deligne-Ribet $q$-expansion principle). More
generally, one can reduce non-commutative main conjectures (for any 
motive) to commutative main conjectures and certain congruences between
special values of $L$-functions of Artin twists of the motive. This is
usually referred to as the strategy of Burns-Kato. I will present a 
formulation of the non-commutative main conjecture and the strategy of 
Burns-Kato. The construction of non-commutative $p$-adic $L$-function and the
proof of non-commutative main conjecture go hand in hand in the Burns-Kato 
strategy. But now we know enough about $K_1$ of Iwasawa algebras to 
construct non-commutative $p$-adic $L$-functions by just proving certain 
congruences (the {\it non-commutative Kummer congruences}) between special 
values of $L$-functions.