We develop the basic theory of curved $$A_{\infty }$$
-categories (
$$cA_{\infty }$$
-categories) in a filtered setting, encompassing the frameworks of Fukaya categories (Fukaya et al. in Part I, AMS/IP studies in advanced mathematics, vol 46, American Mathematical Society, Providence, RI, 2009) and weakly curved $$A_{\infty }$$
-categories in the sense of Positselski (Weakly curved $$A_\infty $$
algebras over a topological local ring, 2012. arxiv:1202.2697v3
). Between two $$cA_{\infty }$$
-categories $$\mathfrak {a}$$
and $$\mathfrak {b}$$
, we introduce a $$cA_{\infty }$$
-category $$\mathsf {qFun}(\mathfrak {a}, \mathfrak {b})$$
of so-called $$qA_{\infty }$$
-functors in which the uncurved objects are precisely the $$cA_{\infty }$$
-functors from $$\mathfrak {a}$$
to $$\mathfrak {b}$$
. The more general $$qA_{\infty }$$
-functors allow us to consider representable modules, a feature which is lost if one restricts attention to $$cA_{\infty }$$
-functors. We formulate a version of the Yoneda Lemma which shows every $$cA_{\infty }$$
-category to be homotopy equivalent to a curved dg category, in analogy with the uncurved situation. We also present a curved version of the bar-cobar adjunction.