Let \(E\) be a Banach space and let \(1\leq p<\infty\). A subset \(K\) of \(E\) is called relatively \(p\)-compact if there exists a sequence \((x_n)_n\in \ell_p(E)\) (\((x_n)_n\in c_0(E)\) if \(p=\infty\)) such that \(K\subset p\)-co\(\{ x_n\}\), where \(p\)-co\(\{ x_n\}=\{\sum^\infty_{n=1}\alpha_nx_n:(\alpha_n)_n\in B_{\ell_{p'}}\}\) and \({1\over p}+{1\over p'}=1\). The \(p\)-compactness is a strengthening of the usual compactness. A bounded linear operator \(T\) from a Banach space \(E\) to a Banach space \(F\) is called \(p\)-compact, written \(T\in {\mathcal{K}}_p(E,F)\), if it maps bounded sets into relatively \(p\)-compact sets. Endowed with a suitable norm, \({\mathcal{K}}_p(E,F)\) is a Banach operator ideal. During the last decade, \(p\)-compactness, related operators, and approximation properties have been studied in rather many papers. But all of them seem to overlook the fact that \(p\)-compactness is a special case of the general \(\mathcal{A}\)-compactness of \textit{B. Carl} and \textit{I. Stephani} [Math. Nachr. 119, 77--95 (1984; Zbl 0575.47015)] which is defined for any operator ideal \(\mathcal{A}\), so that \(p\)-compactness is precisely \({\mathcal{K}}_p\)-compactness. The paper under review discusses the Carl-Stephani theory and uses it to generalize or open previous results on the \(p\)-compactness. In particular, the authors give an alternative proof of the Delgado-Piñeiro-Oja theorem (see [\textit{C. Piñeiro} and \textit{J. M. Delgado}, Proc. Am. Math. Soc. 139, No. 3, 957--967 (2011; Zbl 1270.47018)] and [\textit{E. Oja}, J. Funct. Anal. 263, No. 9, 2876--2892 (2012; Zbl 1301.47030)]) asserting that a sequence \((x_n)_n\subset E\) is \(p\)-null if and only if \((x_n)_n\) is relatively \(p\)-compact and norm convergent to zero. While the Carl-Stephani theory studies \(\mathcal{A}\)-compact sets, \(\mathcal{A}\)-compact operators, \(\mathcal{A}\)-null sequences, etc., for an arbitrary operator ideal \(\mathcal{A}\), the present paper introduces an appropriate topology in the theory and develops it further assuming that \(\mathcal{A}\) is not just an operator ideal, but a Banach operator ideal. The authors introduce a way to measure the size of \(\mathcal{A}\)-compact sets and they use this measure to endow the operator ideal \(\mathcal{K}_{\mathcal{A}}\) of \(\mathcal{A}\)-compact operators with a norm, under which it becomes a Banach operator ideal. They prove that \(\mathcal{K}_{\mathcal{A}}\) is a regular Banach operator ideal and they show that a set is \(\mathcal{A}\)-compact with equal size regardless of whether it is considered as a subset of a Banach space \(E\) or as a subset of its bidual \(E''\). Recall that a Banach space \(E\) has the approximation property if its identity map can be uniformly approximated by finite-rank operators on compact subsets of \(E\). The authors study the weakening of the approximation property, defined by replacing the compact sets by \(\mathcal{A}\)-compact sets. They also study another version of the approximation property, the \(\mathcal{K}_{\mathcal{A}}\)-approximation property, that is a special case of the approximation property defined for an arbitrary Banach operator ideal in [\textit{E. Oja}, J. Math. Anal. Appl. 387, No. 2, 949--952 (2012; Zbl 1241.46013)]. Among others, they prove that the \(\mathcal{K}_{\mathcal{A}}\)-approximation property is implied by the bounded approximation property. Reviewer's remark: Incidentally, the Carl-Stephani theory was at the same time independently rediscovered and applied in [\textit{J. M. Delgado} and \textit{C. Piñeiro}, Stud. Math. 214, No. 1, 67--75 (2013; Zbl 1432.46008)], so that there is a certain overlap with results of the paper under review.