  Generalizing Christensen's notion of a Haar-null set and Darji's notion of a
Haar-meager set, we introduce and study the notion of a Haar-$\mathcal I$ set
in a Polish group. Here $\mathcal I$ is an ideal of subsets of some compact
metrizable space $K$. A Borel subset $B\subset X$ of a Polish group $X$ is
called Haar-$\mathcal I$ if there exists a continuous map $f:K\to X$ such that
$f^{-1}(B+x)\in\mathcal I$ for all $x\in X$. Moreover, $B$ is generically
Haar-$\mathcal I$ if the set of witness functions $\{f\in C(K,X):\forall x\in
X\;\;f^{-1}(B+x)\in\mathcal I\}$ is comeager in the function space $C(K,X)$. We
study (generically) Haar-$\mathcal I$ sets in Polish groups for many concrete
and abstract ideals $\mathcal I$, and construct the corresponding
distinguishing examples. We prove some results on Borel hull of Haar-$\mathcal
I$ sets, generalizing results of Solecki, Elekes, Vidny\'anszky, Dole\v{z}al,
Vlas\v{a}k on Borel hulls of Haar-null and Haar-meager sets. Also we establish
various Steinhaus properties of the families of (generically) Haar-$\mathcal I$
sets in Polish groups for various ideals $\mathcal I$.
