The theory of sparse recovery from compressive measurements relies predominantly on the so-called restricted isometry property. In this talk, I shall summarize recent results based on an appropriate modification of this property. Firstly, I will show how exact recovery of sparse vectors can still be achieved from standard linear measurements. Secondly, I will give a simple explanation for the possibility of approximate recovery when these measurements are quantized to the extreme. Thirdly, I will discuss an intermediate situation where large measurements saturate. I will finally mention analogous results dealing with the recovery of low-rank matrices rather than sparse vectors.