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Fermions with the internal degrees of freedom described in Clifford spacecarry in any dimension a half integer spin. There are two kinds of spins inClifford space. The spin-charge-family theory,assuming even d=13+1, uses onekind of spins to describe in d=3+1 spins and charges of quarks and leptons andantiquarks and antileptons, while the other kind is used to describe families.The new way of second quantization, suggested by the spin-charge-family theory,is presented. It is shown that the creation and annihilation operators of1-fermion states, written as products of nilpotents and projectors of an oddClifford character, fulfill the anticommutation relations as required in thesecond quantization procedure for fermions: 1-fermion states are in Cliffordspace already second quantized, the creation operators for any n-fermion secondquantized vectors are products of one fermion creation operators, operating onthe empty vacuum state. It is demonstrated that also in Grassmann space thereexist the creation and annihilation operators of an odd Grassmann character,generating "fermions", which fulfill as well the anticommutation relations forfermions, representing correspondingly the second quantized 1-"fermion" states,in this case with integer spins. Grassmann space offers no families. We discussthe new second quantization procedure of the fields in both spaces. For theGrassmann case we present the action, basic states, solutions of the Weylequation for free massless "fermions" and discrete symmetry operators. A shortoverview of the achievements of the spin-charge-family theory is done, and openproblems of this theory still waiting to be solved are presented. The Grassmannand the Clifford case are compared in order to better understand open questionsin physics of elementary fermion and boson fields and in cosmology.

New way of second quantized theory of fermions with either Clifford or Grassmann coordinates and spin-charge-family theory