The authors extend the concept of Voronoi diagram in the ordinary Euclidean geometry for n points to the one in the Laguerre geometry for n circles in the plane, where the distance between a circle and a point is defined by the length of the tangent line. Specifically, the distance \(d_ L(C_ i,P)\) between a circle \(C_ i\), with center \((x_ i,y_ i)\) and radius \(r_ i\), and a point \(P=(x,y)\) is defined by \(d^ 2_ L(C_ i,P)=(x-x_ i)^ 2+(y-y_ i)^ 2-r^ 2_ i.\) Then, for n circles \(C_ i\) \((i=1,...,n)\), the Voronoi region \(V(C_ i)\) of circle \(C_ i\) is defined by \(V(C_ i)=\cap_{j}\{point\quad P| d^ 2_ L(C_ i,P)\leq d^ 2_ L(C_ j,P)\}.\) \(V(C_ i)\) \((i=1,...,n)\) partition the plane, which is called the Voronoi diagram in the Laguerre geometry, in short, Laguerre-Voronoi diagram. It is shown that there is an O(n log n) algorithm for this extended case. The Laguerre-Voronoi diagram can be applied to solving effecively a number of geometric problems such as those of determining whether or not a point belongs to the union of n circles, of finding the connected components of n circles, of finding the contour of the union of n circles, and of computing the measure of the union of n circles. By using the Laguerre-Voronoi diagram in three-dimensional space, the volume of n intersecting spheres can be computed in a precise manner. The authors also discuss the Voronoi diagram with respect to the distance of quadratic form. This class of diagrams contains ordinary Euclidean Voronoi diagrams, Euclidean furthest Voronoi diagrams, Laguerre-Voronoi diagrams, section diagrams, etc. It can be shown that every Voronoi diagram in this class can be obtained in linear time from some Laguerre- Voronoi diagram, determined accordingly by that diagram, by means of geometric transforms. This implies that the Voronoi diagram with respect to the distance of quadratic form in the plane can be computed in O(n log n) time.