In this paper we develop state-of-the-art privacy attacks against Large Language Models (LLMs), where an adversary with some access to the model tries to learn something about the underlying training data. Our headline results are new membership inference attacks (MIAs) against pretrained LLMs that perform hundreds of times better than baseline attacks, and a pipeline showing that over 50% (!) of the fine-tuning dataset can be extracted from a fine-tuned LLM in natural settings. We consider varying degrees of access to the underlying model, pretraining and fine-tuning data, and both MIAs and training data extraction. For pretraining data, we propose two new MIAs: a supervised neural network classifier that predicts training data membership on the basis of (dimensionality-reduced) model gradients, as well as a variant of this attack that only requires logit access to the model by leveraging recent model-stealing work on LLMs. To our knowledge this is the first MIA that explicitly incorporates model-stealing information. Both attacks outperform existing black-box baselines, and our supervised attack closes the gap between MIA attack success against LLMs and the strongest known attacks for other machine learning models. In fine-tuning, we find that a simple attack based on the ratio of the loss between the base and fine-tuned models is able to achieve near-perfect MIA performance; we then leverage our MIA to extract a large fraction of the fine-tuning dataset from fine-tuned Pythia and Llama models. Our code is available at github.com/safr-ai-lab/pandora-llm.