We describe the status of our long-term radial velocity (RV) survey of a sample of 1000 GK-giants with the 9.2-m Hobby-Eberly Telescope and its High Resolution Spectrograph, in search for planetary-mass companions. The survey uses the I2 cell calibration method to achieve a 4-6 m s-1 precision of RV measurements, which is quite sufficient, given the observed >20 m s-1 intrinsic RV variability of these stars. The planets discovered so far include two published detections of giant planets around HD17092 and HD102272 and a growing number of plausible candidates currently undergoing the standard verification process. We present a representative sample of RV curves of these stars and describe the line bisector and photometry analyses applied to the data, in order to separate the RV variations generated by orbiting planets from those resulting from stellar phenomena. We also discuss the preliminary characteristics of the GK-giant planets population, which begin to emerge, as its size continues to grow. These include the previously noted apparent existence of a lower limit to orbital radii at about 0.6 AU and a possible lack of correlation between the frequency of occurrence of planets around giants and stellar metallicity, as well as signatures of the dynamical evolution of planets discovered by our survey.