The principle of the Information Bottleneck (Tishby et al., 1999) is to produce a summary of information X optimized to predict some other relevant information Y . In this paper, we propose a novel approach to unsupervised sentence ummarization by mapping the Information Bottleneck principle to a conditional language modelling objective: given a sentence, our approach seeks a compressed sentence that can best predict the next sentence. Our iterative algorithm under the Information Bottleneck objective searches gradually shorter subsequences of the given sentence while maximizing the probability of the next sentence conditioned on the summary. Using only pretrained language models with no direct supervision, our approach can efficiently perform extractive sentence summarization over a large corpus. Building on our unsupervised extractive summarization (BottleSumEx), we then present a new approach to self-supervised abstractive summarization (BottleSumSelf ), where a transformer-based language model is trained on the output summaries of our unsupervised method. Empirical results demonstrate that our extractive method outperforms other unsupervised models on multiple automatic metrics. In addition, we find that our selfsupervised abstractive model outperforms unsupervised baselines (including our own) by human evaluation along multiple attributes.