A total of 156 Granger causal networks of stock markets are constructed by using the Granger causality test and time series sliding window based on stock index data of 34 major stock markets in the world from 2004 to 2017. The topological structures and evolution characteristics of the Granger causal networks are analyzed from the perspective of complex network theory. Empirical results demonstrate that the network topology has a significant difference during the global financial crisis and other periods. The causal relationships among different global stock markets exhibit a jump growth when each major crisis occurs. The contagion path is also short. A causal relationship between any two stock markets can usually be established with one stock market on average, not by using more than five stock markets. For risk contagion, the American stock markets exerted the largest influence in 12 years, followed by the European stock markets. Stock markets with high intermediate contagion ability play an important role in systemic risk contagion. Despite the crucial markets in Europe and America (e.g., USA, Brazil, and Mexico), stock markets with weak network correlation and strong media ability (e.g., the markets of Japan, Korea, Australia, and New Zealand) play a critical role in risk contagion.