A locally compact group G has a maximal compact subgroup if and only if \(G/G_ 0\) has a maximal compact subgroup (Theorem 1). In a totally disconnected locally compact group (such as \(G/G_ 0\) above), every compact subgroup is contained in an open compact subgroup; in particular, maximal compact subgroups are necessarily open. If G is a locally compact group with relatively compact conjugacy classes, and if K is a maximal compact normal subgroup (existing after a result of Bagley and Wu), then G/K is a Lie group. This result remains true for groups with a finite normal subgroup chain such that successive factor groups have this property (Theorem 5). If G is a locally compact group in which the subgroup B(G) of elements with relatively compact conjugacy classes is compactly generated, then G has a maximal compact normal subgroup (Theorem 6). The paper has additional useful results along the same line involving IN- groups, Moore groups, compact extensions of nilpotent normal subgroups.