  This paper is the first in a series in which we offer a new framework for
hermitian K-theory in the realm of stable $\infty$-categories. Our perspective
yields solutions to a variety of classical problems involving Grothendieck-Witt
groups of rings and clarifies the behaviour of these invariants when 2 is not
invertible.
  In this article we lay the foundations of our approach by considering Lurie's
notion of a Poincar\'e $\infty$-category, which permits an abstract counterpart
of unimodular forms called Poincar\'e objects. We analyse the special cases of
hyperbolic and metabolic Poincar\'e objects, and establish a version of
Ranicki's algebraic Thom construction. For derived $\infty$-categories of
rings, we classify all Poincar\'e structures and study in detail the process of
deriving them from classical input, thereby locating the usual setting of forms
over rings within our framework. We also develop the example of visible
Poincar\'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 of
Poincar\'e $\infty$-categories, showing in particular that they form a
bicomplete, closed symmetric monoidal $\infty$-category. We also study the
process of tensoring and cotensoring a Poincar\'e $\infty$-category over a
finite simplicial complex, a construction featuring prominently in the
definition of the L- and Grothendieck-Witt spectra that we consider in the next
instalment.
  Finally, we define already here the 0-th Grothendieck-Witt group of a
Poincar\'e $\infty$-category using generators and relations. We extract its
basic properties, relating it in particular to the 0-th L- and algebraic
K-groups, a relation upgraded in the second instalment to a fibre sequence of
spectra which plays a key role in our applications.
