Tabular data prediction is a fundamental machine learning task for many applications. Existing methods predominantly employ discriminative modeling and operate under the assumption of a fixed target column, necessitating re-training for every new predictive task. Inspired by the generative power of large language models (LLMs), this paper exploits the idea of building universal tabular data predictors based on generative modeling, namely UniPredict. Here, we show that scaling up an LLM to extensive tabular datasets with the capability of comprehending diverse tabular inputs and predicting for target variables following the input instructions. Specifically, we train a single LLM on an aggregation of 169 tabular datasets with diverse targets and compare its performance against baselines that are trained on each dataset separately. We observe this versatile UniPredict model demonstrates an advantage over other models, ranging from 5.4% to 13.4%, when compared with the best tree-boosting baseline and the best neural network baseline, respectively. We further test UniPredict in few-shot learning settings on another 62 tabular datasets. Our method achieves strong performance in quickly adapting to new tasks, where our method outperforms XGBoost over 100% on the low-resource setup and shows a significant margin over all baselines. We envision that UniPredict sheds light on developing a universal tabular data prediction system that learns from data at scale and serves a wide range of prediction tasks.