There is an interesting personal essay on the history and developments in the art and design of numerical schemes to limit spurious oscillations in solutions of nonlinear PDEs even in the presence of shocks here.
I will echo one point that the Sweby diagram (pictured above with cite containing explanation below) universally captures the nature of classical TVD limiters. As a reviewer and associate journal editor I puzzle over the large number of new articles submitted on the topic.
Sweby, Peter K. High resolution schemes using flux limiters for hyperbolic conservation laws. SIAM Journal on Numerical Analysis, Volume 21, Issue 5, 1984, dx.doi link.