  Given a partial Steiner triple system (STS) of order $n$, what is the order
of the smallest complete STS it can be embedded into? The study of this
question goes back more than 40 years. In this paper we answer it for
relatively sparse STSs, showing that given a partial STS of order $n$ with at
most $r \le \varepsilon n^2$ triples, it can always be embedded into a complete
STS of order $n+O(\sqrt{r})$, which is asymptotically optimal. We also obtain
similar results for completions of Latin squares and other designs.
  This suggests a new, natural class of questions, called deficiency problems.
Given a global spanning property $\mathcal{P}$ and a graph $G$, we define the
deficiency of the graph $G$ with respect to the property $\mathcal{P}$ to be
the smallest positive integer $t$ such that the join $G\ast K_t$ has property
$\mathcal{P}$. To illustrate this concept we consider deficiency versions of
some well-studied properties, such as having a $K_k$-decomposition,
Hamiltonicity, having a triangle-factor and having a perfect matching in
hypergraphs.
  The main goal of this paper is to propose a systematic study of these
problems; thus several future research directions are also given.
