This article gives an overview of both theoretical and experimental developments concerning states with lattice symmetry breaking in the cuprate high-temperature superconductors. Recent experiments have provided evidence for states with broken rotation as well as translation symmetry, and will be discussed in terms of nematic and stripe physics. Of particular importance here are results obtained using the techniques of neutron and X-ray scattering and scanning tunnelling spectroscopy. Ideas on the origin of lattice-symmetry-broken states will be reviewed, and effective models accounting for various experimentally observed phenomena will be summarized. These include both weak-coupling and strong-coupling approaches, with a discussion of their distinctions and connections. The collected experimental data indicate that the tendency toward uni-directional stripe-like ordering is common to underdoped cuprates, but becomes weaker with increasing number of adjacent CuO2 layers.