  We present interpretation of known results in the theory of discrete
asymptotic and discrete conjugate nets from the "discretization by B\"{a}cklund
transformations" point of view. We collect both classical formulas of XIXth
century differential geometry of surfaces and their transformations, and more
recent results from geometric theory of integrable discrete equations. We first
present transformations of hyperbolic surfaces within the context of the
Moutard equation and Weingarten congruences. The permutability property of the
transformations provides a way to construct integrable discrete analogs of the
asymptotic nets for such surfaces. Then after presenting the theory of
conjugate nets and their transformations we apply the principle that
B\"{a}cklund transformations provide integrable discretization to obtain known
results on the discrete conjugate nets. The same approach gives, via the
Ribaucour transformations, discrete integrable analogs of orthogonal conjugate
nets.
