This is another of the author’s excellent expository articles. In the first two sections, basic facts about the PWB method for the solution of the Dirichlet problem are recalled, and examples from the theory of parabolic partial differential equations are given to show that domains with irregular boundary points arise very naturally. Thus the scene is set for the main section of the article, in which are described the principal results of three recent papers by the author [Bull. Sci. Math., II. Ser. 109, 337-361 (1985; Zbl 0594.31017); Potential theory and its related fields, Proc. Symp., Kyoto 1986, RIMS Kokyuroku 610, 85-102 (1987; Zbl 0672.31014)] and I. Netuka [Bull. Sci. Math., II. Ser. 114, 1-22 (1990; Zbl 0699.31015)]. These are concerned with the existence of fine limits, and limits along maximal sequences, of bounded harmonic functions at irregular boundary points, in the context of harmonic spaces.