The projective line P(R) over a ring R is the set of all submodules of R2 of type R(a,b), where (a,b) is the first row of some invertible 2×2 matrix over R. If K is a field contained in R as a subring, the generalized chain geometry Σ(K,R) is the geometric structure with point set P(R) and chain set {P(K)g|g∈GL2(R)}. Two points R(a,b) and R(c,d) are said to be distant if (acbd)∈GL2(R). The group GL2(R) acts transitively on the set of all triples of mutually distant points. In Section 2, two examples are described in detail. A blocking set B in Σ(K,R) is a set of points such that every chain contains at least one element of B. The authors give some basic results on the problem of determining the minimum size of blocking sets in a finite chain geometry. Section 3 is concerned with the number λi of chains containing i given mutually distant points (i=0,1,2,3). In Section 4, lower bounds for the size of a blocking set in Σ(K,R) are given, both in the general case and in the case in which R is a local ring. Two examples attaining the general lower bound are exhibited. The authors also construct blocking sets in chain geometries starting from a blocking set in a Möbius geometry.