The counting class $\text{#P}$ and the related decision class $PP$ both involve counting the number of certificates to $NP$ problems.

Because $\text{#P}$ counts certificates, it seems obvious that $NP \subseteq \text{#P}$ and $co-NP \subseteq \text{#P}$. Furthermore, we can find any certificates using a brute force search, so $\text{#P} \subseteq EXP$.

The most interesting result that I found in Arora and Barak is Toda's Theorem which states that $PH \subseteq P^{\text{#SAT}}$, $\text{#SAT}$ being a $\text{#P}$-complete problem.

I'm wondering if there are any other results which relate $\text{#P}$ or $PP$ to other complexity classes, and what relationships are conjectured. For example, is it conjectured or known that $\text{#P} \subsetneq PSPACE$ ?

It occurred to me that because $\text{#P}$ is not a decision class some of these relationship may not be well-defined, but then again it seems natural enough to wonder how much time and space it takes to compute non-boolean (i.e. non-decision) functions of languages. Arora and Barak call Toda's Theorem a "big surprise", since as they put it $\text{#P}$ and $PH$ "both are natural generalizations of $NP$, but it seemed that their features ... are not directly comparable to each other". Given that result I'm hopeful that $\text{#P}$ can be related to other classes.