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Unique twistor--like Lorentz harmonic formulation for all N=1 supersymmetricextended objects (super-p-branes) moving in the space--time of arbitrarydimension D (admissible for given $p$) are suggested. The equations of motionare derived, explicit form of the \kappa-symmetry transformations is presentedand the classical equivalence to the standard formulation is proved.  The cases with minimal world--sheet dimensions $p=1,2$, namely of D=10heterotic string and D=11 supermembrane, are considered in details. Inparticular, the explicit form of irreducible \kappa-symmetry transformationsfor D=11 supermembrane is derived.Comment: (revized: minor changes, which was nesessary because previous version  cannot be read LATEX byond Padova University), 24 pages, LATEX ,  DEPD/94/TH/3

N=1 super- p -branes in twistor-like Lorentz harmonic formulation