As usual, Abelian-by-finite (nilpotent-by-finite) groups are called AF-groups (NF-groups). The authors prove: (1) If the group \(G\) is neither perfect nor an AF-group (NF-group) but all proper subgroups are AF-groups (NF-groups), then \(G\) is periodic (Theorems 2.1, 2.5); (2) If the group \(G\) is perfect, does not possess infinite simple images, and all its proper subgroups are NF-groups, then \(G\) is hyperabelian and countable, it is the join of an ascending sequence of nilpotent normal subgroups, and all proper subgroups are nilpotent and ascendent. Finitely generated subgroups of \(G\) are subnormal in \(G\) (Proposition 3.1). Non-perfect non-locally-nilpotent non-NF-groups with all proper subgroups NF are characterized (Corollary 2.7).