Manually segmenting multiple sclerosis (MS) cortical lesions (CLs) is extremely time consuming, and past studies have shown only moderate inter-rater reliability. To accelerate this task, we developed a deep-learning-based framework (CLAIMS: Cortical Lesion AI-Based Assessment in Multiple Sclerosis) for the automated detection and classification of MS CLs with 7 T MRI. Two 7 T datasets, acquired at different sites, were considered. The first consisted of 60 scans that include 0.5 mm isotropic MP2RAGE acquired four times (MP2RAGEx4), 0.7 mm MP2RAGE, 0.5 mm T-2*-weighted GRE, and 0.5 mm T-2*-weighted EPI. The second dataset consisted of 20 scans including only 0.75 x 0.75 x 0.9 mm(3) MP2RAGE. CLAIMS was first evaluated using sixfold cross-validation with single and multi-contrast 0.5 mm MRI input. Second, the performance of the model was tested on 0.7 mm MP2RAGE images after training with either 0.5 mm MP2RAGEx4, 0.7 mm MP2RAGE, or alternating the two. Third, its generalizability was evaluated on the second external dataset and compared with a state-of-the-art technique based on partial volume estimation and topological constraints (MSLAST). CLAIMS trained only with MP2RAGEx4 achieved results comparable to those of the multi-contrast model, reaching a CL true positive rate of 74% with a false positive rate of 30%. Detection rate was excellent for leukocortical and subpial lesions (83%, and 70%, respectively), whereas it reached 53% for intracortical lesions. The correlation between disability measures and CL count was similar for manual and CLAIMS lesion counts. Applying a domain-scanner adaptation approach and testing CLAIMS on the second dataset, the performance was superior to MSLAST when considering a minimum lesion volume of 6 mu L (lesion-wise detection rate of 71% versus 48%). The proposed framework outperforms previous state-of-the-art methods for automated CL detection across scanners and protocols. In the future, CLAIMS may be useful to support clinical decisions at 7 T MRI, especially in the field of diagnosis and differential diagnosis of MS patients.