  A path-integral quantization method is proposed for dynamical systems whose
classical equations of motion do \textit{not} necessarily follow from the
action principle. The key new notion behind this quantization scheme is the
Lagrange structure which is more general than the Lagrangian formalism in the
same sense as Poisson geometry is more general than the symplectic one. The
Lagrange structure is shown to admit a natural BRST description which is used
to construct an AKSZ-type topological sigma-model. The dynamics of this
sigma-model in $d+1$ dimensions, being localized on the boundary, are proved to
be equivalent to the original theory in $d$ dimensions. As the topological
sigma-model has a well defined action, it is path-integral quantized in the
usual way that results in quantization of the original (not necessarily
Lagrangian) theory. When the original equations of motion come from the action
principle, the standard BV path-integral is explicitly deduced from the
proposed quantization scheme. The general quantization scheme is exemplified by
several models including the ones whose classical dynamics are not variational.
