  The N = 3 string models are special solutions of the type II perturbative
string theories. We present explicit expressions for the helicity
supertraces,which count the number of the perturbative BPS multiplets. Assuming
the non-perturbative duality $(S\leftrightarrow T)$ of the heterotic string on
T^6 and type II on K_3 x T^2 valid in N = 4 theories, we derive the N = 3
non-perturbative BPS mass formula by "switching off" some of the N = 4 charges
and "fixing" to special values some of the N = 4 moduli. This operation
corresponds to a well-defined Z_2 projection acting freely on the
compactification manifold. The consistency of this projection and the precise
connection of the N = 4 and N = 3 BPS spectrum is shown explicitly in several
type II string constructions. The heterotic N = 3 and some asymmetric type II
constructions turn out to be non-perturbative with the S moduli fixed at the
self-dual point S=i. Some of the non-perturbative N = 3 type II are defined in
the context of F-theory. The bosonic sector of the N=3 string effective action
is also presented. This part can be useful for the study of 4d black holes in
connection with the asymptotic density of BPS states in N = 3 string theory.
