We review recent muon spin rotation and relaxation (μSR) studies performed on insulating magnetic systems crystallizing in the pyrochlore structure and for which the only magnetic species are rare-earth ions. Different points are discussed: attempts to measure directly the magnetic charge of the monopoles in classical spin-ice systems, the frequency shift of Tb₂Ti₂O₇ at 20 mK which is at variance with expectation for a quantum spin-ice system, and the detection of unexpected short-range magnetic correlations in the magnetically ordered state of Yb₂Ti₂O₇, Yb₂Sn₂O₇, and Er₂Ti₂O₇. An explanation for the ubiquitous persistent spin dynamics is given in terms of spin loops. Their description requires to go beyond mean-field, i.e., at least Gaussian fluctuations have to be taken into account. While in the magnetically ordered state a spontaneous field is detected for some systems, such as Nd₂Sn₂O₇, it is absent for others, such as Yb₂Sn₂O₇. A theoretical work suggests this feature to be related to the magnetic structure adopted by the magnetic moments. A μSR work on praseodymium-based pyrochlore compounds is also discussed. Normal rare-earth spinel systems for which the magnetic ions form a network of corner sharing regular tetrahedra as in the pyrochlores are finally mentioned.