In this talk, we will discuss a dyadic variant of the Hilbert transform, which is a useful model of its continuous counterpart and the prototypical example of a so-called "Haar shift". After discussing some background and motivation in the Lebesgue measure case, we will turn to the situation where the L^2 Haar functions are defined with respect to a locally finite Borel measure μ, which may not satisfy the dyadic doubling condition. In this more general setting, Lopez-Sanchez, Martell, and Parcet identified a weak regularity condition on the measure μ which characterizes weak-type and L^p estimates for this dyadic Hilbert transform. I then will discuss joint work with Jose Conde Alonso and Jill Pipher, where we obtain a domination of the dyadic Hilbert transform (and more generally, Haar shifts) by a modified sparse form. As an application, we characterize the class of weights where the dyadic Hilbert transform and related operators are bounded. A surprising novelty is that the usual (dyadic) Muckenhoupt A_2 condition is necessary, but no longer sufficient in the non-doubling setting, and our modified weight condition reflects the "complexity" of the underlying Haar shift. Finally, we will examine a different dyadic Haar shift model of the Hilbert transform and its relationship to BMO (bounded mean oscillation) functions via commutators in the non-doubling setting (joint with Tainara Borges, Jose Conde Alonso, and Jill Pipher).