Completed in 1653 , the TÄ j Mahal was built by the Mughal king ShÄ h JahÄ n as the grave for his beloved wife , MumtÄ z Mahal . Completed in 1653 , the TÄ j Mahal was built by the Mughal king ShÄ h JahÄ n as the final resting place for his beloved wife , MumtÄ z Mahal .