The direct growth of large-area nanosheets of diverse phase-pure Ruddlesden-Popper perovskites enables the fabrication of arbitrary vertical heterostructures and multi-heterostructures of perovskites. Ruddlesden-Popper lead halide perovskites have emerged as a new class of two-dimensional semiconductors with tunable optoelectronic properties, potentially offering unlimited heterostructure configurations for exploration. However, the practical realization of such heterostructures is challenging because of the difficulty in achieving controllable direct synthesis or van der Waals integration of halide perovskites due to their mobile and fragile crystal lattices. Here we report direct growth of large-area nanosheets of diverse phase-pure Ruddlesden-Popper perovskites with thicknesses down to one monolayer at the solution-air interface and a reliable approach for gently transferring and stacking these nanosheets. These advances enable the deterministic fabrication of arbitrary vertical heterostructures and multi-heterostructures of Ruddlesden-Popper perovskites with greater structural degrees of freedom that define the electronic structures of the heterojunctions. Such rationally designed heterostructures exhibit interesting interlayer properties, such as interlayer carrier transfer and reduction of the photoluminescence linewidth, and could enable the exploration of exciton physics and optoelectronic applications.