  Given two Banach spaces $X$ and $Y$, we analyze when the projective tensor
product $X\widehat{\otimes}_\pi Y$ has Corson's property (C) or is weakly
Lindel\"of determined (WLD), subspace of a weakly compactly generated (WCG)
space or subspace of a Hilbert generated space. For instance, we show that: (i)
$X\widehat{\otimes}_\pi Y$ is WLD if and only if both $X$ and $Y$ are WLD and
all operators from $X$ to $Y^*$ and from $Y$ to $X^*$ have separable range;
(ii) $X\widehat{\otimes}_\pi Y$ is subspace of a WCG space if the same holds
for both $X$ and $Y$ under the assumption that every operator from $X$ to $Y^*$
is compact; (iii) $\ell_p(\Gamma)\widehat{\otimes}_\pi \ell_q(\Gamma)$ is
subspace of a Hilbert generated space for any $1< p,q<\infty$ such that
$1/p+1/q<1$ and for any infinite set $\Gamma$. We also pay attention to the
injective tensor product $X\widehat{\otimes}_\varepsilon Y$. In this case, the
stability of property (C) and the property of being WLD turn out to be closely
related to the condition that all regular Borel probability measures on the
dual ball have countable Maharam type. Along this way, we generalize a result
of Plebanek and Sobota that if $K$ is a compact space such that $C(K\times K)$
has property (C), then all regular Borel probability measures on $K$ have
countable Maharam type. This generalization provides a consistent negative
answer to a question of Ruess and Werner about the preservation of the
$w^*$-angelicity of the dual unit ball under injective tensor products.
