IceCube is a cubic-kilometer Cherenkov detector in the deep ice at the geographic South Pole. The dominant event yield in the deep ice detector consists of penetrating atmospheric muons with energies above approximately 300 GeV, produced in cosmic ray air showers. In addition, the surface array, IceTop, measures the electromagnetic component and GeV muons of air showers. Hence, IceCube and IceTop yield unique opportunities to study cosmic rays with unprecedented statistics in great detail. We will present recent results of comic ray measurements from IceCube and IceTop. In this overview, we will highlight measurements of the energy spectrum of cosmic rays from 250 TeV up to the EeV range and their mass composition above 3 PeV. We will also report recent results from measurements of the muon content in air showers and discuss their consistency with predictions from current hadronic interaction models.