The field of thermoelectrics has progressed enormously and is now growing steadily because of recently demonstrated advances and strong global demand for cost-effective, pollution-free forms of energy conversion. Rapid growth and exciting innovative breakthroughs in the field over the last 10-15 years have occurred in large part due to a new fundamental focus on nanostructured materials. As a result of the greatly increased research activity in this field, a substantial amount of new data--especially related to materials--have been generated. Although this has led to stronger insight and understanding of thermoelectric principles, it has also resulted in misconceptions and misunderstanding about some fundamental issues. This article sets out to summarize and clarify the current understanding in this field; explain the underpinnings of breakthroughs reported in the past decade; and provide a critical review of various concepts and experimental results related to nanostructured thermoelectrics. We believe recent achievements in the field augur great possibilities for thermoelectric power generation and cooling, and discuss future paths forward that build on these exciting nanostructuring concepts.