  We define a monodromy homomorphism for irreducible families of regular
elliptic fibrations which takes values in the mapping class group of a
punctured sphere. We compute the monodromy for elliptic fibrations only which
contain no singular fibres of types other than I_1 and I*_0. There are maximal
groups, which can be the monodromy groups of algebraic, resp. differentiable
families of elliptic surfaces, and we give an algebraic criterion for the
equality of both groups which we can check to apply in case the number of I_1
fibres is at most 6.
