Adding visual cues to audio-based speech separation can improve separation performance. This paper introduces AV-CrossNet, an \gls{av} system for speech enhancement, target speaker extraction, and multi-talker speaker separation. AV-CrossNet is extended from the CrossNet architecture, which is a recently proposed network that performs complex spectral mapping for speech separation by leveraging global attention and positional encoding. To effectively utilize visual cues, the proposed system incorporates pre-extracted visual embeddings and employs a visual encoder comprising temporal convolutional layers. Audio and visual features are fused in an early fusion layer before feeding to AV-CrossNet blocks. We evaluate AV-CrossNet on multiple datasets, including LRS, VoxCeleb, and COG-MHEAR challenge. Evaluation results demonstrate that AV-CrossNet advances the state-of-the-art performance in all audiovisual tasks, even on untrained and mismatched datasets.