Without the axiom of choice, cardinalities of sets may not be wellordered or linearly ordered. Under the axiom of determinacy, $\mathsf{AD}$, and its extensions, the cardinality below many familiar sets can be distinguished and shown to have an interesting natural structure under injections. We will discuss some recent results concerning cardinalities of sets below the power set of certain cardinals under determinacy assumptions. Cardinalities of partition spaces (for example, $[\kappa]^\epsilon$ and $[\kappa]^{<\kappa}$) will be distinguished by suitable partition properties and continuity of functions on partition spaces. The perfect set property completely characterizes $\mathcal{P}(\omega)$. We will discuss some recent results about the structure of the cardinalities below $\mathcal{P}(\omega_1)$ under various determinacy assumptions. For instance, we will discuss the complete structure of the cardinalities below ${}^\omega\omega_1$ (due to Woodin under $\mathsf{AD}_{\mathbb{R}}$ and $\mathsf{DC}$) and how to extend this result to sets of $\omega$-sequences through higher uncountable cardinals under $\mathsf{AD}_{\frac{1}{2}\mathbb{R}}$. We will also discuss how the structure of cardinalities below $\mathcal{P}(\omega_1)$ is different in $L(\mathbb{R})$ (where uniformization fails). This is joint work with Stephen Jackson and Nam Trang.