Set Theory without Powerset I'm looking for references for a set theory which does not include the axiom of powerset but includes axioms allowing taking cartesian product. Please refer me to such if you know about any.
 A: Elaborating on the comments:
First of all, even within the classical $\mathsf{ZFC}$-style context, powerset-free theories play an important role. The example I know most about is $\mathsf{KP}$, which basically consists of set theory without choice or powerset, and with separation and replacement dramatically restricted to "simple" formulas. An ordinal $\alpha$ such that $L_\alpha\models\mathsf{KP}$ is called admissible, and admissible ordinals occur not just in set theory but also in computability theory and proof theory. There is also "$\mathsf{ZFC}$ without powerset" (the details of which are more complicated than one might guess); this theory doesn't have the same significance outside set theory that $\mathsf{KP}$ does, but within set theory it too is quite important. Basically, one way we analyze a given "big" set theory is by looking at how "small" pieces of its models behave. Such truncated models are especially interesting when they satisfy a "strong-but-not-too-strong" theory, since then they occur frequently but still have some decent closure properties. Both $\mathsf{KP}$ and $\mathsf{ZFC}^-$ (not to be confused with $\mathsf{ZFC-}$, per the above-linked article) are such a happy-medium theories in the right context, but there are many others.
Outside the classical context there are additional examples but to my knowledge they don't enjoy nearly the same prominence as the $\mathsf{ZFC}$ fragments mentioned above. The SEP article on alternative set theories mentions two, pocket set theory (which Asaf Karagila brought up above) and an alternative due to Vopenka. Each has the property that there is a maximal cardinality. Note that this alone does not negate powerset - consider e.g. $\mathsf{NFU}$ or positive set theories, which in fact have a universal set.
