The Compact Linear Collider (CLIC) is a mature option for a future electron-positron collider operating at centre-of-mass energies of up to 3TeV. The accelerator uses a two-beam acceleration scheme, in which normal-conducting high-gradient accelerating structures are powered via a high-current drive beam. CLIC will be built and operated in a staged approach with three centre-of-mass energy stages currently assumed to be 380GeV, 1.5TeV, and 3TeV. The detector concept matches the physics performance requirements and the CLIC experimental conditions. The initial energy stage of CLIC will focus on precision measurements of Higgs-boson and top-quark properties. The subsequent energy stages enhance the reach of many searches for Beyond Standard Model physics and give access to the Higgs self-coupling with a precision of around 10%. A selection of results from recent studies will be presented showing that CLIC has excellent sensitivity to many BSM physics scenarios, both through direct observation and precision measurements of SM processes. New particles can be discovered in a model-independent way almost up to the kinematic limit. Compared with hadron colliders, the low background conditions at CLIC provide extended discovery potential, also for non-standard signatures such as charged long-lived particles. In addition to studying new particles directly, BSM models can be probed up to scales of tens of TeV through precision measurements.