Residual intersections of licci ideals behave well in a sense that residual intersection of ideals of 2x2 minors of a 2xn matrix, for n at least 4, do not. Recently Craig Huneke and Bernd Ulrich and I have been studying a way in which these residual intersections, and a certain wider class of ideals whose powers have linear presentations do behave well after all. I'll review the background and discuss what we've learned.