In this notes, we survey the recent developments on theory of generalized quasidisks. Based on the more or less standard techniques used earlier, we also provide some minor improvements on the recorded results. A few nature questions were posed.2010 Mathematics Subject Classification. 30C62,30C65. Key words and phrases. homeomorphism of finite distortion, generalized quasidisk, local connectivity, three point property, cusps.C.-Y.

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Abstract. For s > 0 given, we consider a planar domain Ω s with a rectifiable boundary but containing a cusp of degree s, and show that there is no homeomorphism f : R 2 → R 2 of finite distortion with exp(λK) ∈ L 1 loc (R 2 ) so that f (B) = Ω s when λ > 4/s and B is the unit disc. On the other hand, for λ < 2/s such an f exists. The critical value for λ remains open.

Abstract. We give sharp integrability conditions on the distortion of a planar homeomorphism that maps a standard cusp onto the unit disk.

Abstract. We consider planar homeomorphisms f : R 2 → R 2 that are of finite distortion and map the unit disk onto a specific cusp domain Ω s . We study the relation between the degree s of the cusp and the integrability of the distortion function K f by sharpening a previous result where K f is assumed to be locally exponentially integrable.