In this talk for fixed p,1 < p <∞,we consider a measure, u^, associated with a positivep harmonic function u^  defined in an open set G in R² and vanishing on a portion of the boundary of G. During the first part of the talk we outline results obtained by the presenter and coauthors when G is a bounded simply connected domainin R² and u^ is p harmonic nearthe boundary of G with continuous boundary value zero on the boundary of G. Our results generalize a famous the orem of Makarov concerning the dimension of harmonic measure. During the second part of this talk we consider p harmonic measure in Rⁿ, n >3: In this case we outline some examples constructed by the presenter and coauthors. We also discuss recent work by the presenter and coauthors on a conjecture of his concerning the ` natural generalization ' of a theorem of Jones and Wolf for harmonic measure to the p harmonic setting when p≥n. Finally given time we discuss work in progress with coauthors.