Large cardinals are useful in set theory in part because they imply certain reflection properties. For example, if $\kappa$ is a weakly compact cardinal, then $\kappa$ satisfies the tree property and every stationary subset of $\kappa$ reflects. An interesting direction of research involves investigating the extent to which these reflection properties can hold at smaller cardinals. By results of Levy and Solovay, for most large cardinal notions (inaccessible, weakly compact, measurable, etc.), if $\kappa$ is such a large cardinal in $V$ and $\mathbb{P}$ is a forcing poset such that $|\mathbb{P}| < \kappa$, then $\kappa$ remains large in the forcing extension $V^{\mathbb{P}}$. Therefore, reflection properties of a cardinal $\kappa$ that are implied by $\kappa$ being a particular large cardinal are themselves indestructible by small forcing when $\kappa$ is such a large cardinal. However, these reflection properties may no longer be indestructible at $\kappa$ if $\kappa$ is a small cardinal. For example, it is consistent that $\aleph_{\omega+1}$ has the tree property but there is $\mathbb{P}$ with $|\mathbb{P}| = \aleph_1$ such that $\aleph_{\omega+1}$ fails to have the tree property in $V^{\mathbb{P}}$.

We will begin by reviewing the relevant large cardinal and reflection notions and will then consider strengthenings of certain reflection properties that are always indestructible under small forcing, focusing in particular on stationary reflection and the tree property. We will look at the extent to which these reflection properties can hold at small cardinals and the extent to which they are in fact stronger than the weaker principles from which they are derived.