You may wish to look at http://plato.stanford.edu/entries/qt-measurement/ (Ref. A) to get a better idea of the quantum measurement problem.
Let me say a few words about the article by Allahverdyan, Balian, Nieuwenhuizen (ABN) (it is published in Physics Reports, but you can also find a version of it at http://arxiv.org/abs/1107.2138 ). I should note that I am no expert on quantum measurements, so please take the following for what it's worth.
I think ABN's work is outstanding, and this is not just my opinion. I hear from different people that this is the best we have now on this issue. I remember I attended Balian's presentation a few years ago, and Scully (an author of quantum theory of laser) commented (I cannot be sure about the exact words): "Good work. Why did not I do it?"
You probably know that standard quantum mechanics typically includes two parts: unitary evolution (say, the Schroedinger equation) and the theory of quantum measurements (say, the collapse postulate). These two parts are, strictly speaking, mutually contradictory (please see Ref. A). Using a specific model of quantum measurement, ABN derive the theory of quantum measurements from unitary evolution. Of course, they could not derive it rigorously, as you cannot mathematically derive a conclusion that contradicts the assumption, but they derive it as a physical approximation valid under some conditions. Their model describes a measurement of a spin projection of a particle using a measurement apparatus containing a large number of spins interacting with a phonon bath. The spin system is initially in a metastable paramagnetic state. Due to interaction with the particle, this system transitions to a ferromagnetic state with a lower energy. The resulting magnetization reflects the particle spin projection. You may conclude from ABN's results that the theory of quantum measurements is not an independent part of quantum theory, but an approximate consequence of unitary evolution.
ABN claim a solution of the measurement problem, and they may be right in some sense. I would like to add though that their work is not only outstanding, it is also highly disruptive (in the same sense as a business idea can be disruptive). The side effect of their work is that the theory of quantum measurements is just an approximation in the best case, the collapse postulate and even the Born rule are just approximations. For example, strictly speaking, you cannot even have a unique outcome of measurement: while the spin system transitions to a ferromagnetic state, this state is not final, and the system will return to the paramagnetic state due to Poincare's reversal, although this will take enormous time. Another consequence of ABN's results: measurement is independent of an observer - the result is registered permanently (modulo extremely slow reversal).
You ask in your comment: "Does this have to relate with Einsteins, hidden variables?" I think so, as, for example, in the Bell theorem, you have to use both unitary evolution and the theory of quantum measurements to prove that the inequalities can be violated in standard quantum theory. And if the theory of quantum measurements is just an approximation, all bets are off, if you ask me, as I cannot imagine what "approximate nonlocality" can possibly be. Furthermore, ABN emphasize that registration of a measurement outcome is a relatively slow process, as the apparatus must be a macroscopic system, and this may be relevant to the locality loophole in Bell experiments.