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The most common approach to speaker diarization is clustering of speaker embeddings. However, the clustering-based approach has a number of problems; i.e., (i) it is not optimized to minimize diarization errors directly, (ii) it cannot handle speaker overlaps correctly, and (iii) it has trouble adapting their speaker embedding models to real audio recordings with speaker overlaps. To solve these problems, we propose the End-to-End Neural Diarization (EEND), in which a neural network directly outputs speaker diarization results given a multi-speaker recording. To realize such an end-to-end model, we formulate the speaker diarization problem as a multi-label classification problem and introduce a permutation-free objective function to directly minimize diarization errors. Besides its end-to-end simplicity, the EEND method can explicitly handle speaker overlaps during training and inference. Just by feeding multi-speaker recordings with corresponding speaker segment labels, our model can be easily adapted to real conversations. We evaluated our method on simulated speech mixtures and real conversation datasets. The results showed that the EEND method outperformed the state-of-the-art x-vector clustering-based method, while it correctly handled speaker overlaps. We explored the neural network architecture for the EEND method, and found that the self-attention-based neural network was the key to achieving excellent performance. In contrast to conditioning the network only on its previous and next hidden states, as is done using bidirectional long short-term memory (BLSTM), self-attention is directly conditioned on all the frames. By visualizing the attention weights, we show that self-attention captures global speaker characteristics in addition to local speech activity dynamics, making it especially suitable for dealing with the speaker diarization problem.