Strong high-frequency excitation (HFE) may change the ‘slow’ (i.e. effective or average) properties of mechanical systems, e.g. their stiffness, natural frequencies, equilibriums, equilibrium stability, and bifurcation paths. This tutorial describes three general HFE effects: Stiffening – an apparent change in the stiffness associated with an equilibrium; Biasing – a tendency for a system to move towards a particular state which does not exist or is unstable without HFE; and Smoothening – a tendency for discontinuities to be apparently smeared out by HFE. The effects and a method for analyzing them are introduced first in terms of simple physical examples, and then in generalized form for mathematical models covering broad classes of discrete and continuous mechanical systems. Several application examples are summarized. Three mathematical tools for analyzing HFE effects are described and compared: The Method of Direct Separation of Motions, the Method of Averaging, and the Method of Multiple Scales. The tutorial concludes by suggesting that more vibration experts, researchers and students should know about HFE effects, for the benefit not only of general vibration troubleshooting, but also for furthering the creation of innovative technical devices and processes utilizing HFE effects.