Rising demand, climate change, growing fuel costs, outdated power system infrastructures, and new power generation technologies have made renewable distribution generators very attractive in recent years. Because of the increasing penetration level of renewable energy sources in addition to the growth of new electrical demand sectors, such as electrical vehicles, the power system may face serious problems and challenges in the near future. A revolutionary new power grid system, called smart grid, has been developed as a solution to these problems. The smart grid, equipped with modern communication and computation infrastructures, can coordinate different parts of the power system to enhance energy efficiency, reliability, and quality, while decreasing the energy cost. Since conventional distribution networks lack smart infrastructures, much research has been recently done in the distribution part of the smart grid, called smart distribution grid (SDG). This paper surveys contemporary literature in SDG from the perspective of the electricity market in addition to power system considerations. For this purpose, this paper reviews current demand side management methods, supply side management methods, and electrical vehicle charging and discharging techniques in SDG and also discusses their drawbacks. We also present future research directions to tackle new and existing challenges in the SDG.