What are some (non image) datasets where deep networks are required for good performance?
Welcome to the community!

In text-related tasks, DL have shown amazing breakthrough e.g. in NLU for reading comprehension or question answering. If you have heard about Google's BERT, that was one of big steps which, to some extent, pushed the boundaries of SOTA in NLP/NLU last year and currently, a heavy wave is going on doing research on different NLP/NLU tasks mostly based on either the idea coming from BERT or directly from its pretrained models.

The quora QA dataset, SQuAD dataset, etc. are great starting points to play with bert and its successors and see how great the idea was.

Hope it helps!