This paper reports on the learning of difficult and complex concepts, the characteristics of these concepts that make them difficult for students to learn and understand well, the kinds of misconceptions students acquire, and the difficulty of changing these mistaken beliefs. A scheme, the Conceptual Stability Scheme, is presented for analyzing a concept and its related concepts for potential stability. A set of studies is reviewed that confirmed that the misconception that was predicted to be highly stable according to the Conceptual Stability Scheme was, in fact, highly stable. The mental maneuvers (knowledge shields) learners engage in to ward off changing their beliefs are presented and discussed. Many different kinds of shields are used in a variety of circumstances. (Contains 23 figures and 48 references.) (Author/SLD)