  All our observations that characterise space and time are expressed in terms
of non-local, bi-tensorial objects such as geodesic intervals between events
and two-point (Green) functions. In this contribution, I highlight the
importance of characterising spacetime geometry in terms of such non-local
objects, focusing particularly on two important bi-tensors that play a
particular fundamental role -- Synge's World function and the van Vleck
determinant. I will first discuss how these bi-tensors help capture information
about spacetime geometry, and then describe their role in characterising
quantum spacetime endowed with a lower bound, say $\ell_0$, on spacetime
intervals. Incorporating such a length scale in a Lorentz covariant manner
necessitates a description of spacetime geometry in terms of above bi-tensors,
and naturally replaces the conventional description based on the metric tensor
$g_{ab}(x)$ with a description in terms of a non-local bi-tensor $q_{ab}(x,
y)$. The non-analytic structure of $q_{ab}(x, y)$ which renders a perturbative
expansion in $\ell_0$ meaningless, also generically leaves a non-trivial
``relic" in the limit $\ell_0 \to 0$. I present some results where such a relic
term is manifest; specifically, I will discuss how this: (i) suggests a
description of gravitational dynamics different from the one based on
Einstein-Hilbert lagrangian, (ii) implies dimensional reduction to $2$ at small
scales, (iii) connects with the notion of cosmological constant itself being a
non-local vestige of the small scale structure of spacetime, (iv) helps address
the issues of spacetime singularities. I will conclude by discussing the
ramifications of these ideas for quantum gravity.
