Large language models have demonstrated remarkable capabilities across a wide range of tasks, yet their ability to process structured symbolic knowledge remains underexplored. To address this gap, we propose a taxonomy of ontological capabilities and introduce OntoURL, the first comprehensive benchmark designed to systematically evaluate LLMs' capabilities in handling ontologies -- formal and symbolic representations of domain knowledge. Based on the proposed taxonomy, OntoURL systematically assesses three dimensions: understanding, reasoning, and learning through 15 distinct tasks comprising 57,303 questions derived from 40 ontologies across 8 domains. Experiments with 20 open-source LLMs reveal significant performance differences across models, tasks, and domains, with current LLMs showing capabilities in understanding ontological knowledge but weaknesses in reasoning and learning tasks. Further experiments with few-shot and chain-of-thought prompting illustrate how different prompting strategies affect model performance. Additionally, a human evaluation reveals that LLMs outperform humans in understanding and reasoning tasks but fall short in most learning tasks. These findings highlight both the potential and limitations of LLMs in processing symbolic knowledge and establish OntoURL as a critical benchmark for advancing the integration of LLMs with formal knowledge representations.