  In the Steiner Tree Augmentation Problem (STAP), we are given a graph $G =
(V,E)$, a set of terminals $R \subseteq V$, and a Steiner tree $T$ spanning
$R$. The edges $L := E \setminus E(T)$ are called links and have non-negative
costs. The goal is to augment $T$ by adding a minimum cost set of links, so
that there are 2 edge-disjoint paths between each pair of vertices in $R$. This
problem is a special case of the Survivable Network Design Problem, which can
be approximated to within a factor of 2 using iterative rounding~\cite{J2001}.
  We give the first polynomial time algorithm for STAP with approximation ratio
better than 2. In particular, we achieve an approximation ratio of $(1.5 +
\varepsilon)$. To do this, we employ the Local Search approach of~\cite{TZ2022}
for the Tree Augmentation Problem and generalize their main decomposition
theorem from links (of size two) to hyper-links.
  We also consider the Node-Weighted Steiner Tree Augmentation Problem
(NW-STAP) in which the non-terminal nodes have non-negative costs. We seek a
cheapest subset $S \subseteq V \setminus R$ so that $G[R \cup S]$ is
2-edge-connected. Using a result of Nutov~\cite{N2010}, there exists an $O(\log
|R|)$-approximation for this problem. We provide an $O(\log^2
(|R|))$-approximation algorithm for NW-STAP using a greedy algorithm leveraging
the spider decomposition of optimal solutions.
