Estimators of the finite population covariance with several systems of weights are considered. New calibrated estimators of the finite population covariance (variance) are derived, using two and three weighting systems that are defined by various calibration equations and loss functions. The expressions of approximate variance for some of these estimators are presented. The estimators derived are compared by simulation. Finally, it is shown how the calibrated estimators of the covariance may be applied in regression estimation of the finite population total.