Models pretrained with self-supervised objectives on large text corpora achieve state-of-the-art performance on English text summarization tasks. However, these models are typically fine-tuned on hundreds of thousands of data points, an infeasible requirement when applying summarization to new, niche domains. In this work, we introduce a novel and generalizable method, called WikiTransfer, for fine-tuning pretrained models for summarization in an unsupervised, dataset-specific manner. WikiTransfer fine-tunes pretrained models on pseudo-summaries, produced from generic Wikipedia data, which contain characteristics of the target dataset, such as the length and level of abstraction of the desired summaries. WikiTransfer models achieve state-of-the-art, zero-shot abstractive summarization performance on the CNN-DailyMail dataset and demonstrate the effectiveness of our approach on three additional diverse datasets. These models are more robust to noisy data and also achieve better or comparable few-shot performance using 10 and 100 training examples when compared to few-shot transfer from other summarization datasets. To further boost performance, we employ data augmentation via round-trip translation as well as introduce a regularization term for improved few-shot transfer. To understand the role of dataset aspects in transfer performance and the quality of the resulting output summaries, we further study the effect of the components of our unsupervised fine-tuning data and analyze few-shot performance using both automatic and human evaluation.