Evaluating the open-ended outputs of large language models (LLMs) has become a bottleneck as model capabilities, task diversity, and modality coverage rapidly expand. Existing "LLM-as-a-Judge" evaluators are typically narrow in a few tasks, aspects, or modalities, and easily suffer from low consistency. In this paper, we argue that explicit, fine-grained aspect specification is the key to both generalizability and objectivity in automated evaluation. To do so, we introduce a hierarchical aspect taxonomy spanning 112 aspects that unifies evaluation across four representative settings - Natural Language Generation, Image Understanding, Image Generation, and Interleaved Text-and-Image Generation. Building on this taxonomy, we create FRAbench, a benchmark comprising 60.4k pairwise samples with 325k aspect-level labels obtained from a combination of human and LLM annotations. FRAbench provides the first large-scale, multi-modal resource for training and meta-evaluating fine-grained LMM judges. Leveraging FRAbench, we develop GenEval, a fine-grained evaluator generalizable across tasks and modalities. Experiments show that GenEval (i) attains high agreement with GPT-4o and expert annotators, (ii) transfers robustly to unseen tasks and modalities, and (iii) reveals systematic weaknesses of current LMMs on evaluation.