Privacy policies are often obfuscated by their complexity, which impedes transparency and informed consent. Conventional machine learning approaches for automatically analyzing these policies demand significant resources and substantial domain-specific training, causing adaptability issues. Moreover, they depend on extensive datasets that may require regular maintenance due to changing privacy concerns. In this paper, we propose, apply, and assess PAPEL (Privacy Policy Analysis through Prompt Engineering for LLMs), a framework harnessing the power of Large Language Models (LLMs) through prompt engineering to automate the analysis of privacy policies. PAPEL aims to streamline the extraction, annotation, and summarization of information from these policies, enhancing their accessibility and comprehensibility without requiring additional model training. By integrating zero-shot, one-shot, and few-shot learning approaches and the chain-of-thought prompting in creating predefined prompts and prompt templates, PAPEL guides LLMs to efficiently dissect, interpret, and synthesize the critical aspects of privacy policies into user-friendly summaries. We demonstrate the effectiveness of PAPEL with two applications: (i) annotation and (ii) contradiction analysis. We assess the ability of several LLaMa and GPT models to identify and articulate data handling practices, offering insights comparable to existing automated analysis approaches while reducing training efforts and increasing the adaptability to new analytical needs. The experiments demonstrate that the LLMs PAPEL utilizes (LLaMA and Chat GPT models) achieve robust performance in privacy policy annotation, with F1 scores reaching 0.8 and above (using the OPP-115 gold standard), underscoring the effectiveness of simpler prompts across various advanced language models.