  We analyze the landscape of general smooth Gaussian functions on the sphere
in dimension $N$, when $N$ is large. We give an explicit formula for the
asymptotic complexity of the mean number of critical points of finite and
diverging index at any level of energy and for the mean Euler characteristic of
level sets. We then find two possible scenarios for the bottom landscape, one
that has a layered structure of critical values and a strong correlation
between indexes and critical values and another where even at levels below the
limiting ground state energy the mean number of local minima is exponentially
large. We end the paper by discussing how these results can be interpreted in
the language of spin glasses models.
