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The breakthrough in neural networks starting with AlexNet in 2012 is that they can represent semantic meaning with a vector of numbers. This can be used to group similar ideas, recognize positive or negative sentiment even with sarcasm, and convert ideas between representations. That includes translating from English to French, slang to formal writing, or images to text.

The "generative" ones are trained to predict the next token given a sequence. But that's not the only use for language models or large neural networks.