TITLE: Status of Beilinson conjectures?
QUESTION [12 upvotes]: (I hesitate to post that question here, but I received on answer on FB:) 
Does anyone know how the current status of work on them is? And how the possible generalizations etc. which one thinks currently on, look like?

REPLY [10 votes]: In addition to Andreas's excellent answer, we should also mention the Tamagawa number conjecture of Bloch and Kato, which predicts the undetermined rational factor arising in Beilison's conjectural description of the $L$-value. The Bloch-Kato conjecture was later reformulated and generalized by Fontaine and Perrin-Riou to the case of motives with coefficients in an arbitrary number field. Here are some references :
Bloch, Kato, L-functions and Tamagawa numbers of motives.
Fontaine, Perrin-Riou, Autour des conjectures de Bloch et Kato: cohomologie galoisienne et
 valeurs de fonctions L.
Colmez, Fonctions L p-adiques.
Kings, The Bloch-Kato conjecture on special values of L-functions. A survey of known results.
Flach, The equivariant Tamagawa number conjecture : A survey.
Gealy, On the Tamagawa Number Conjecture for Motives Attached to Modular Forms.
Bellaïche, An introduction to the conjecture of Bloch and Kato.
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