Four versions of compactness (equivalent in ZFC) and their properties are investigated without the axiom of choice. All the four versions are equivalent iff the axiom of choice holds (many equivalent forms concerning mainly products of spaces are given). The Boolean prime ideal theorem holds iff the compactness defined by open covers is equivalent to compactness defined by convergence of ultrafilters (or to compactness of completely regular spaces defined as closed subspaces of Tychonov cubes). The last case and some more other cases are again accompanied by several equivalent forms for products or compactifications.