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The requirement of ${\cal N}=1$ supersymmetry for M-theory backgrounds of theform of a warped product ${\cal M}\times_{w}X$, where $X$ is an eight-manifoldand ${\cal M}$ is three-dimensional Minkowski or AdS space, implies theexistence of a nowhere-vanishing Majorana spinor $\xi$ on $X$. $\xi$ lifts to anowhere-vanishing spinor on the auxiliary nine-manifold $Y:=X\times S^1$, where$S^1$ is a circle of constant radius, implying the reduction of the structuregroup of $Y$ to $Spin(7)$. In general, however, there is no reduction of thestructure group of $X$ itself. This situation can be described in the languageof generalized $Spin(7)$ structures, defined in terms of certain spinors of$Spin(TY\oplus T^*Y)$. We express the condition for ${\cal N}=1$ supersymmetryin terms of differential equations for these spinors. In an equivalentformulation, working locally in the vicinity of any point in $X$ in terms of a`preferred' $Spin(7)$ structure, we show that the requirement of ${\cal N}=1$supersymmetry amounts to solving for the intrinsic torsion and all irreducibleflux components, except for the one lying in the $\bf{27}$ of $Spin(7)$, interms of the warp factor and a one-form $L$ on $X$ (not necessarilynowhere-vanishing) constructed as a $\xi$ bilinear; in addition, $L$ isconstrained to satisfy a pair of differential equations. The formalism based onthe group $Spin(7)$ is the most suitable language in which to describesupersymmetric compactifications on eight-manifolds of $Spin(7)$ structure,and/or small-flux perturbations around supersymmetric compactifications onmanifolds of $Spin(7)$ holonomy.Comment: 24 pages. V2: introduction slightly extended, typos corrected in the  text, references added. V3: the role of Spin(7) clarified, erroneous  statements thereof corrected. New material on generalized Spin(7) structures  in nine dimensions. To appear in JHE

M-theory on eight-manifolds revisited: Script N = 1 supersymmetry and generalized Spin(7) structures