This study tests for the presence of linear and nonlinear dependences in returns and volatility for six agricultural futures daily prices series, three traded on MATIF Euronext (wheat, corn, and rapeseed), and three traded on Chicago Board of Trade (red winter wheat, corn, and soybean) over the period 2000-2013. Whereas price dynamics on the Chicago Board of Trade (CBOT) market seem to fit classical GARCH modelling, time series dependences in the MATIF market cannot be fully described by short-term dependences alone. According to various criteria, the results suggest the presence of long memories for the European market. However, the low fractional order of ARFIMA-type or FiGARCH-type models can explain only some, but not all, of the observed nonlinearity. Nonlinearity could be influenced by the regime shift. By taking volatility breaks in the series into account, it is possible to gain a better understanding of the serial dependencies.