Multimodal positron emission tomography-computed tomography (PET-CT) is used routinely in the assessment of cancer. PET-CT combines the high sensitivity for tumor detection with PET and anatomical information from CT. Tumor segmentation is a critical element of PET-CT but at present, there is not an accurate automated segmentation method. Segmentation tends to be done manually by different imaging experts and it is labor-intensive and prone to errors and inconsistency. Previous automated segmentation methods largely focused on fusing information that is extracted separately from the PET and CT modalities, with the underlying assumption that each modality contains complementary information. However, these methods do not fully exploit the high PET tumor sensitivity that can guide the segmentation. We introduce a multimodal spatial attention module (MSAM) that automatically learns to emphasize regions (spatial areas) related to tumors and suppress normal regions with physiologic high-uptake. The resulting spatial attention maps are subsequently employed to target a convolutional neural network (CNN) for segmentation of areas with higher tumor likelihood. Our MSAM can be applied to common backbone architectures and trained end-to-end. Our experimental results on two clinical PET-CT datasets of non-small cell lung cancer (NSCLC) and soft tissue sarcoma (STS) validate the effectiveness of the MSAM in these different cancer types. We show that our MSAM, with a conventional U-Net backbone, surpasses the state-of-the-art lung tumor segmentation approach by a margin of 7.6% in Dice similarity coefficient (DSC).