In this talk we recall the notion of characteristic p invariants of a ring (commutative and Noetherian), namely the Hilbert-Kunz (HK) function and Hilbert-Kunz multiplicity, introduced by P. Monsky in the 1980s. Here we concentrate on HK multiplicity, which is a more subtle invariant of a ring (compared to classical multiplicity): It is related to characteristic p features of the singularities of the ring. Here we will give an overview of results and some known computations of the HK multiplicity. In higher dimensions one proves that, though Frobenius semistability does not behave well reduction mod p, an invariant associated to the Frobenius instability degree of a bundle V, namely \mu_(HK)(V)-a_(HK)(V), which goes to 0 as p goes to infinity.