Self-organized criticality is considered a common phenomenon in nature and became a fascinating research subject for neuroscience, when critical avalanches were predicted theoretically and observed experimentally to occur in networks of neurons. I will discuss different mechanism to archive self-organization toward criticality in neuronal networks. I will also discuss the problems arising when we try to judge whether an observed system is critical or not. One of the problems is posed by the finite (rather small) size of the system in recordings as well as in simulations leading to a cutoff. Another is connected to unavoidable subsampling that may distort the experimental results. So far, closeness of avalanche size distribution to power law and closeness of branching ratio to one were mainly used, but in finite systems considered so far distribution is neither truly a power law, nor have a constant branching ratio. I will present the model that overcomes the finite size effects in a stochastic branching model by adding a rapid activity dependent adaptation. This truly critical model serves then as a reference point to test hypotheses about criticality. In the rest of the time I would speak about stable self-organization in the balanced neural networks by synaptic plasticities. I will present simulation work as well as analytical understanding of network functioning.