  Part I. We prove a one-to-one correspondence between differential symmetry
breaking operators for equivariant vector bundles over two homogeneous spaces
and certain homomorphisms for representations of two Lie algebras, in
connection with branching problems of the restriction of representations. We
develop a new method (F-method) based on the algebraic Fourier transform for
generalized Verma modules, which characterizes differential symmetry breaking
operators by means of certain systems of partial differential equations. In
contrast to the setting of real flag varieties, continuous symmetry breaking
operators of Hermitian symmetric spaces are proved to be differential operators
in the holomorphic setting. In this case symmetry breaking operators are
characterized by differential equations of second order via the F-method.
  Part II. Rankin--Cohen brackets are symmetry breaking operators for the
tensor product of two holomorphic discrete series representations of SL(2,R).
We address a general problem to find explicit formulae for such intertwining
operatorsin the setting of multiplicity-free branching laws for reductive
symmetric pairs. We discover explicit formulae of new differential symmetry
breaking operators for all the six different complex geometries arising from
semisimple symmetric pairs of split rank one, and reveal an intrinsic reason
why the coefficients of orthogonal polynomials appear in these operators
(Rankin--Cohen type) in the three geometries and why normal derivatives are
symmetry breaking operators in the other three cases. Further, we analyze a new
phenomenon that the multiplicities in the branching laws of Verma modules may
jump up at singular parameters.
