In this article, we explore a new model with three parameters called Gamma-Gompertz (GGo) distribution which is generated by the gamma-X family. The GGo distribution provides better fits than some competing distributions such as three-parameter Gompertz, Gompertz-Lindley, Gompertz-geometric, Gompertz-poisson, and inverse Gompertz distributions. Some well-known distributions are included in the GGo distribution as special sub-models. The density function of the distribution can be decreasing, unimodal and decreasing-increasing-decreasing shaped while the failure rate function can be increasing and unimodal shaped. Various properties of the GGo distribution are obtained. An extensive simulation study is carried out to assess the effectiveness of some classical estimation approaches which are discussed to estimate the model parameters. To demonstrate the potentiality of the GGo distribution, two real data sets are used and the bootstrap percentile confidence intervals are also obtained by bootstrap resampling. In addition, a new regression model is proposed via re-parametrization of the GGo distribution, called the log-GGo distribution. The maximum likelihood method is considered to estimate the unknown parameters of re-parametrized log-GGo distribution. The potentiality of log-GGo regression model is analyzed for HIV+ censored data set.