  In this paper we define multiple Dedekind zeta values (MDZV), using a new
type of iterated integrals, called iterated integrals on a membrane. One should
consider MDZV as a number theoretic generalization of Euler's multiple zeta
values. Over imaginary quadratic fields MDZV capture, in particular, multiple
Eisenstein series (Gangl, Kaneko and Zagier). We give an analogue of multiple
Eisenstein series over real quadratic field and an alternative definition of
values of multiple Eisenstein-Kronecker series (Goncharov). Each of them is a
special case of multiple Dedekind zeta values. MDZV are interpolated into
functions that we call multiple Dedekind zeta functions (MDZF). We show that
MDZF have integral representation, can be written as infinite sum, and have
analytic continuation. We compute explicitly the value of a multiple residue of
certain MDZF over a quadratic number field at the point (1,1,1,1). Based on
such computations, we state two conjectures about MDZV.
