  We explore Two Higgs Doublet Models with non-standard flavor structures. In
analogy to the four, well studied, models with natural flavor conservation
(type 1, type 2, lepton-specific, flipped), we identify four models that
preserve an approximate $U(2)^5$ flavor symmetry acting on the first two
generations. In all four models, the couplings of the 125 GeV Higgs are
modified in characteristic flavor non-universal ways. The heavy neutral and
charged Higgs bosons show an interesting non-standard phenomenology. We discuss
their production and decay modes and identify the most sensitive search
channels at the LHC. We study the effects on low energy flavor violating
processes finding relevant constraints from $B_d$ and $B_s$ meson oscillations
and from the rare decay $B_s \to \mu^+ \mu^-$. We also find that lepton flavor
violating $B$ meson decays like $B_s \to \tau \mu$ and $B \to K^{(*)} \tau \mu$
might have branching ratios at an observable level.
