TITLE: State of the progess of the automated proof checking
QUESTION [20 upvotes]: I recently came across a concept of automated proof checking. I am very intrigued by the idea, that in the future all the proofs could be verified by a computer.
Moreover, some proofs were already verified. However, as I understand those proofs were formulated within the first-order logic.
In layman’s terms could somebody explain to the rest of the mathematicians: how complicated proofs can we verify already? What is the current progress on verifying major results, what are the challenges and possible approaches?
Thank you.

REPLY [9 votes]: Some complicated proofs have been verified by computer, the well known example in mathematics are:

the 4-colors theorem by Georges Gonthier and Benjamin Werner.
the Feit-Thompson theorem by Gy Georges Gonthier and its team.
the Kepler conjecture (Tomas Hales' theorem) by the flyspeck team. 

There are other examples in Computer Science:

a C compiler: CompCert by Xavier Leroy
the functional correctness of a general-purpose operating system kernel by NICTA

As Henning explained the proof theory has already been well explored and it is quite clear how to check a formal proof. 
But there are still many challenges:

how to exchange proofs between different systems
how to edit proof and check proof in wikipedia like environment
what is the best way to formalize some given concept
how to modify the proof assistant to allow  a language closer to the usual mathematical language: overloaded notations, implicit steps in the proof, implicit coercions, how to deal with proofs by analogy or symmetry, ...
how to mix efficient computations and proofs
how to automate proofs
...