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

Bootstrap and Amplitudes: A Hike in the Landscape of Quantum Field Theory