Recurrent neural networks (RNNs) and long short-term memory (LSTM) networks can process sequential data like text and time series data. RNNs have memory and can perform the same task for every element in a sequence, but struggle with long-term dependencies. LSTMs address this issue using memory cells and gates that allow them to learn long-term dependencies. LSTMs have four interacting layers - a forget gate, input gate, cell state, and output gate that allow them to store and access information over long periods of time. RNNs and LSTMs are applied to tasks like language modeling, machine translation, speech recognition, and image caption generation.