Informally, a continuous self-map $f$ on a compact metric space $X$ has the shadowing property provided that behaviors witnessed by the pseudo-orbits of a system (i.e. orbits with some allowed amount of error) are representative of true behaviors of the system in the sense that every pseudo-orbit has an orbit which approximates it.

Surprisingly, despite being quite a strong property and having connections to many other dynamical properties, shadowing has been shown to be a generic property of continuous self-maps for certain classes of spaces.

Motivated by this, in this talk we examine the set $\mathcal T(X)$ of maps with shadowing as a subset of $\mathcal C(X)$, the space of continuous self-maps on a compact metric space $X$. We will discuss the structure of $\mathcal T(X)$ for certain classes of spaces, with a special focus on the question of whether $\mathcal T(X)$ is a generic set in $\mathcal C(X)$.