The Qing Dynasty faced a number of serious problems during this century, problems that would lead to its collapse early in the 20th century.
Perhaps the biggest challenge was from the outside. This started with the British forcing China to open up to the opium trade in the Opium Wars. By the end of the century, European countries had claimed all sorts of concessions and extraterritorial rights in Chinese coastal cities like Shanghai.
The inability of the Qing to resist the West helped lead to another challenge. This was a challenge from within the country. One manifestation of this was the Boxer Rebellion just after the 19th century ended. The Qing were also having serious problems trying to improve the economic and social conditions of the country. This weakness led to the Taiping Rebellion in the 1850s and '60s.
Thus, the Qing Dynasty faced challenges from within and from foreign countries.