The problem of estimation of the common location parameter of two exponential distributions when the scale parameters are unknown and possibly unequal is considered. Three different estimators, namely, the maximum likelihood estimator (MLE), the uniformly minimum variance unbiases estimator (UMVUE) and a modified MLE, are proposed. Their performances are compared in terms of the mean squared error criterion. The biases of the MLE and the modified MLE are also compared. The case when the ratio of the scale parameters is known is also taken into account, and the uniformly minimum variance unbiased estimator of the common location parameter is proposed;Next, the more general problem of estimating the common location parameter of several exponentials with unknown and possibly unequal scale parameters is addressed. Estimators similar to the ones mentioned in the preceding paragraph are proposed, and their performances are compared in terms of biases and mean squared errors;Finally, the problem of estimating the location parameter of an exponential with known coefficient of variation is considered. The best (in the sense of minimum mean squared error) linear combination of the sample minimum and the sample sum of deviations from the minimum is obtained. This estimator is shown to be dominated by the best scale invariant estimator. Also, a class of Bayes estimators is proposed, and the best scale invariant estimator is shown to be a limiting Bayes estimator.