  We define and study the notion of a locally bounded enriched category over a
(locally bounded) symmetric monoidal closed category, generalizing the locally
bounded ordinary categories of Freyd and Kelly. In addition to proving several
general results for constructing examples of locally bounded enriched
categories and locally bounded closed categories, we demonstrate that locally
bounded enriched categories admit fully enriched analogues of many of the
convenient results enjoyed by locally bounded ordinary categories. In
particular, we prove full enrichments of Freyd and Kelly's reflectivity and
local boundedness results for orthogonal subcategories and categories of models
for sketches and theories. We also provide characterization results for locally
bounded enriched categories in terms of enriched presheaf categories, and we
show that locally bounded enriched categories admit useful adjoint functor
theorems and a representability theorem. We also define and study the notion of
$\alpha$-bounded-small weighted limit enriched in a locally $\alpha$-bounded
closed category, which parallels Kelly's notion of $\alpha$-small weighted
limit enriched in a locally $\alpha$-presentable closed category, and we show
that enriched categories of models of $\alpha$-bounded-small weighted limit
theories are locally $\alpha$-bounded.
