Dictionary learning has emerged as a powerful method for data-driven extraction of features from data. The initial focus was from an algorithmic perspective, but recently there has been increasing interest in the theoretical underpinnings. These rely on information-theoretic analytic tools and help us understand the fundamental limitations of dictionary-learning algorithms. We focus on theoretical aspects and summarize results on dictionary learning from vector- and tensor-valued data. Results are stated in terms of lower and upper bounds on sample complexity of dictionary learning, defined as the number of samples needed to identify or reconstruct the true dictionary underlying data from noiseless or noisy samples, respectively. Many analytic tools that help yield these results come from information theory, including restating the dictionary-learning problem as a channel-coding problem and connecting analysis of minimax risk in statistical estimation to Fano’s inequality. In addition to highlighting effects of parameters on the sample complexity of dictionary learning, we show the potential advantages of dictionary learning from tensor data and present unaddressed problems.