The sum of the absolute values of the eigenvalues of a graph is called the energy of the graph. We study the problem of finding graphs with extremal energy within specified sets of graphs. We develop some tools for treating such problems and obtain some partial results. In particular, we show that in many cases the expected extremal graphs with a small number of distinct eigenvalues do not exist and that actual extremal graphs could have a large number of distinct eigenvalues. Zigzag and central circuit structure