  This article is an introduction to two currently very active research
programs, the Conformal Bootstrap and Scattering Amplitudes. Rather than
attempting full surveys, the emphasis is on common ideas and methods shared by
these two seemingly very different programs. In both fields, mathematical and
physical constraints are placed directly on the physical observables in order
to explore the landscape of possible consistent quantum field theories. We give
explicit examples from both programs: the reader can expect to encounter
boiling water, ferromagnets, pion scattering, and emergent symmetries on this
journey into the landscape of local relativistic quantum field theories. The
first part is written for a general physics audience. The second part includes
further details, including a new on-shell bottom-up reconstruction of the
$\mathbb{CP}^1$ model with the Fubini-Study metric arising from re-summation of
the $n$-point interaction terms derived from amplitudes.
