  We establish a correspondence between certain Lie superalgebras with 16
fermionic generators, and half-BPS solutions to supergravities with 32
supersymmetries. Three cases are discussed. For Type IIB supergravity, we
relate semi-simple Lie superalgebras H with 16 fermionic generators which are
subalgebras of PSU(2,2|4), to families of half-BPS solutions which are
invariant under H, and locally asymptotic to AdS_5 x S^5. Similarly, for
M-theory, we relate semi-simple Lie superalgebras H with 16 fermionic
generators which are subalgebras of OSp(8^*|4) or OSp(8|4,R) to families of
half-BPS solutions which are invariant under H, and locally asymptotic to AdS_7
x S^4 or AdS_4 x S^7 respectively. Possible enhancements to more than 16
supersymmetries, such as 24, are also analyzed. The classification of
semi-simple subalgebras of PSU(2,2|4), OSp(8^*|4), and OSp(8|4,R) derived in
this paper, leads us to conjecture the existence of various new families of
half-BPS solutions to Type IIB supergravity and M-theory.
