  We discuss a recently proposed method of quantizing general non-Lagrangian
gauge theories. The method can be implemented in many different ways, in
particular, it can employ a conversion procedure that turns an original
non-Lagrangian field theory in $d$ dimensions into an equivalent Lagrangian
topological field theory in $d+1$ dimensions. The method involves, besides the
classical equations of motion, one more geometric ingredient called the
Lagrange anchor. Different Lagrange anchors result in different quantizations
of one and the same classical theory. Given the classical equations of motion
and Lagrange anchor as input data, a new procedure, called the augmentation, is
proposed to quantize non-Lagrangian dynamics. Within the augmentation
procedure, the originally non-Lagrangian theory is absorbed by a wider
Lagrangian theory on the same space-time manifold. The augmented theory is not
generally equivalent to the original one as it has more physical degrees of
freedom than the original theory. However, the extra degrees of freedom are
factorized out in a certain regular way both at classical and quantum levels.
The general techniques are exemplified by quantizing two non-Lagrangian models
of physical interest.
