The spatio temporal evolution of electromagnetic (Weibel-type) and electrostati c (two-stream) instabilities of electron beams propagating in collisionle ss plasmas were investigated by using electromagnetic relativistic particl e-in-cell simulations. The onset and nonlinear saturation of Weibel instability , the emergence of beams and magnetic field reconnection were discussed. T he excitation of longitudinal electric field plays an important role for the stable propagation of electron beams, which can be destructed by the transverse modulation due to two-stream instability. These results may be important for understanding the static magnetic field generation and fast electron propagati on in fast ignition physics.