We present efficient data structures for problems on unit discs and arcs of their boundary in the plane. (i) We give an output-sensitive algorithm for the dynamic maintenance of the union of *n* unit discs under insertions in *O(k log ^{2} n)* update time and

*O(n)*space, where

*k*is the combinatorial complexity of the structural change in the union due to the insertion of the new disc. (ii) As part of the solution of (i) we devise a fully dynamic data structure for the maintenance of lower envelopes of pseudo-lines, which we believe is of independent interest. The structure has

*O(log*update time and

^{2}n)*O(log n)*vertical ray shooting query time. To achieve this performance, we devise a new algorithm for finding the intersection between two lower envelopes of pseudo-lines in

*O(log n)*time, using

*tentative*binary search; the lower envelopes are special in that at \em>x=-∞ any pseudo-line contributing to the first envelope lies below every pseudo-line contributing to the second envelope. (iii) We also present a dynamic range searching structure for a set of circular arcs of unit radius (not necessarily on the boundary of the union of the corresponding discs), where the ranges are unit discs, with

*O(n log n)*preprocessing time,

*O(n*query time and

^{1/2+ε}+ l)*O(log*amortized update time, where

^{2}n)*l*is the size of the output and for any

*ε>0*. The structure requires

*O(n)*storage space.