This objective was abandoned when technological problems meant that the US would reach the moon first, and Soyuz then developed into a variety of projects (both military and civilian), mostly in conjunction with space stations.
As of 2003, Soyuz derivatives provide Russia's human spaceflight capability and are used to ferry personnel and supplies to and from the International Space Station
Soyuz missions include:
Vostok program | Voskhod program | Soyuz program |