Liver cancer is among the most prevalent and life-threatening cancers worldwide, with hepatocellular carcinoma and liver metastases contributing significantly to cancer related mortality. Accurate segmentation of liver tumors in abdominal computed tomography and magnetic resonance imaging is crucial for tumor prediction, radiation therapy, and treatment monitoring. In recent years, medical image segmentation has advanced rapidly due to deep learning (DL), particularly convolutional neural networks (CNN) and transformer-based architectures. This paper provides a comprehensive review of DL based liver tumor segmentation methods in abdominal imaging. We examine the evolution from traditional encoder decoder frameworks such as U-Net and V-Net to residual, attention based, and hybrid CNN transformer models, including vision transformers. The review also analyzes performance across publicly available benchmarks, including the Medical Segmentation Decathlon, 3D-IRCADb, CHAOS, and the Liver Tumor Segmentation Challenge, using evaluation metrics such as Dice Similarity Coefficient, Hausdorff Distance, and Intersection over Union. Furthermore, we discuss key challenges including tumor heterogeneity, class imbalance, limited annotated data, and domain adaptation across imaging protocols. Emerging directions such as foundation models, self-supervised learning, federated learning, and clinically deployable AI segmentation systems are also highlighted.