  We present a time-resolved spectral analysis of 51 long and 11 short bright
GRBs observed with the {\em Femri}/GBM, paying special attention to $E_{\rm p}$
evolution within a same burst. Among 8 single-pulse long GRBs, 5 show
hard-to-soft evolution, while 3 show intensity-tracking. The multi-pulse long
GRBs have more complicated patterns. Among the GRBs whose time-resolved
spectrum is available for the first pulse, almost half (15/32 GRBs) show clear
hard-to-soft evolution, and the other half (17/32 GRBs) show clear
intensity-tracking. Later pulses typically show the tracking behavior, although
a hard-to-soft evolution pattern was identified in the 2nd pulse of 2 GRBs
whose pulses are well separated. Statistically, the hard-to-soft evolution
pulses tend to be more asymmetric than the intensity-tracking ones, with a
steeper rising wing than the falling wing. Short GRBs have $E_{\rm p}$ tracking
intensity exclusively with the 16ms time resolution analysis. We performed a
simulation analysis, and suggest that at least for some bursts, the late
intensity-tracking pulses could be a consequence of overlapping hard-to-soft
pulses. However, the fact that the intensity-tracking pattern exists in the
first pulse of multi-pulse long GRBs and some single-pulse GRBs suggest that
intensity tracking is an independent component, which may operate in some late
pulses as well. For the GRBs with measured redshifts, we present a
time-resolved $E_{\rm p}-L_{\gamma, \rm iso}$ correlation analysis and show
that the scatter of the correlation is comparable to that of the global
Amati/Yonetoku relation. We discuss the predictions of various radiation models
regarding $E_{\rm p}$ evolution, as well as the possibility of a precession jet
in GRBs. It seems that the data pose great challenge to all these models, and
hold the key to unveil the physics of GRB prompt emission.
