  We demonstrate and develop dyadic-probabilistic methods in connection with
non-homogeneous bilinear operators, namely singular integrals and square
functions. We develop the full non-homogeneous theory of bilinear singular
integrals using a modern point of view. The main result is a new global $Tb$
theorem for Calder\'on-Zygmund operators in this setting. Our main tools
include maximal truncations, adapted Cotlar type inequalities and suppression
and big piece methods.
  While proving our bilinear results we also advance and refine the linear
theory of Calder\'on-Zygmund operators by improving techniques and results. For
example, we simplify and make more efficient some non-homogeneous summing
arguments appearing in $T1$ type proofs. As a byproduct, we can manage with
ease quite general modulus of continuity in the kernel estimates. Our testing
conditions are also quite general by virtue of the big piece method of proof.
