Treatment with broadly neutralizing antibodies (bNAbs) has proven effective against HIV-1 infections in humanized mice, non-human primates, and humans. Due to the high mutation rate of HIV-1, resistance testing of the patients viral strains to the bNAbs is still inevitable. So far, bNAb resistance can only be tested in expensive and time-consuming neutralization experiments. Here, we introduce well-performing computational models that predict the neutralization response of HIV-1 to bNAbs given only the envelope sequence of the virus. Using non-linear support vector machines based on a string kernel, the models learnt even the important binding sites of bNAbs with more complex epitopes, i.e., the CD4 binding site targeting bNAbs, proving thereby the biological relevance of the models. To increase the interpretability of the models, we additionally provide a new kind of motif logo for each query sequence, visualizing those residues of the test sequence that influenced the prediction outcome the most. Moreover, we predicted the neutralization sensitivity of around 34,000 HIV-1 samples from different time points to a broad range of bNAbs, enabling the first analysis of HIV resistance to bNAbs on a global scale. The analysis showed for many of the bNAbs a trend towards antibody resistance over time, which had previously only been discovered for a small non-representative subset of the global HIV-1 population.