We introduce a generalization of the O(N) field theory to N-colored membran
es of arbitrary inner dimension D. The O(N) model is obtained for D --> 1,
while N --> 0 leads to self-avoiding tethered membranes (as the O(N) model
reduces to self-avoiding polymers). The model is studied perturbatively by
a 1-loop renormalization group analysis, and exactly as N --> infinity. Fre
edom to choose the expansion point D, leads to precise estimates of critica
l exponents of the O(N) model. Insights gained from this generalization inc
lude a conjecture on the nature of droplets dominating the 3d-Ising model a
t criticality; and the fixed point governing the random bond Ising model.