  We find bounds on scalar masses resulting from a criterion of naturalness, in
a broad class of two Higgs doublet models (2HDMs). Specifically, we assume the
cancellation of quadratic divergences in what are called the type I, type II,
lepton-specific and flipped 2HDMs, with an additional U(1) symmetry. This
results in a set of relations among masses of the physical scalars and coupling
constants, a generalization of the Veltman conditions of the Standard Model.
Assuming that the lighter $CP$-even neutral Higgs particle is the observed
scalar particle of mass $\sim$125 GeV, and imposing further the constraints
from the electroweak T-parameter, stability, and perturbative unitarity, we
calculate the range of the mass of each of the remaining physical scalars.
