The usual supermembrane solution of D = 11 supergravity interpolates b
etween R(11) and AdS(4) x round S-7, has symmetry P-3 x SO(8) and pres
erves 1/2 of the spacetime supersymmetries for either orientation of t
he round S-7. Here we show that more general supermembrane solutions m
ay be obtained by replacing the round S-7 by any seven-dimensional Ein
stein space M(7). These have symmetry P-3 X G, where G is the isometry
group of M(7). For example, G = SO(5) x SO(3) for the squashed S-7. F
or one orientation of M(7), they preserve N/16 spacetime supersymmetri
es where 1 less than or equal to N less than or equal to 8 is the numb
er of Killing spinors on M(7); for the opposite orientation they prese
rve no supersymmetries since then M(7) has no Killing spinors. For exa
mple N = 1 for the left-squashed S-7 owing to its G(2) Weyl holonomy,
whereas N = 0 for the right-squashed S-7. All these solutions saturate
the same Bogomol'nyi bound between the mass and charge. Similar repla
cements of S-D-p-2 by Einstein spaces M(D-p-2) yield new super p-brane
solutions in other spacetime dimensions D less than or equal to 11. I
n particular, simultaneous dimensional reduction of the above D = 11 s
upermembranes on S-1 leads to a new class of D = 10 elementary string
solutions which also have fewer supersymmetries.