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This paper is the first in a series in which we offer a new framework forhermitian K-theory in the realm of stable $\infty$-categories. Our perspectiveyields solutions to a variety of classical problems involving Grothendieck-Wittgroups of rings and clarifies the behaviour of these invariants when 2 is notinvertible.  In this article we lay the foundations of our approach by considering Lurie'snotion of a Poincar\'e $\infty$-category, which permits an abstract counterpartof unimodular forms called Poincar\'e objects. We analyse the special cases ofhyperbolic and metabolic Poincar\'e objects, and establish a version ofRanicki's algebraic Thom construction. For derived $\infty$-categories ofrings, we classify all Poincar\'e structures and study in detail the process ofderiving them from classical input, thereby locating the usual setting of formsover rings within our framework. We also develop the example of visiblePoincar\'e structures on $\infty$-categories of parametrised spectra,recovering the visible signature of a Poincar\'e duality space.  We conduct a thorough investigation of the global structural properties ofPoincar\'e $\infty$-categories, showing in particular that they form abicomplete, closed symmetric monoidal $\infty$-category. We also study theprocess of tensoring and cotensoring a Poincar\'e $\infty$-category over afinite simplicial complex, a construction featuring prominently in thedefinition of the L- and Grothendieck-Witt spectra that we consider in the nextinstalment.  Finally, we define already here the 0-th Grothendieck-Witt group of aPoincar\'e $\infty$-category using generators and relations. We extract itsbasic properties, relating it in particular to the 0-th L- and algebraicK-groups, a relation upgraded in the second instalment to a fibre sequence ofspectra which plays a key role in our applications.

Hermitian K-theory for stable $\infty$-categories I: Foundations