Nanopore sensing is a key technology for single-molecule detection and analysis. Solid-state nanopores have emerged as a versatile platform, since their fabrication allows to engineer their properties by controlling size, shape, and chemical functionalization. However, lithography-based fabrication approaches for non-planar nanopores-on-chip rely on polymers that have limits with respect to hard- and robustness, durability, and refractive index. In this respect, nanopores made of metal oxides with high dielectric constant would be much more favourable and have the potential to extend the suitability of solid-state nanopores towards optoelectronic technologies. Here, we present a versatile method to fabricate three-dimensional nanopores of different dielectric oxides with controlled shapes. Our approach uses photoresist only as a template in the focused-ion-beam lithography to define the nanopore shape, which is subsequently coated with different oxides (SiO2, Al2O3, TiO2 and HfO2) by atomic-layer deposition. Then the photoresist is fully removed by chemo-physical treatment, resulting in nanopores entirely made from dielectric oxides on a thin solid-state membrane. Our methodology allows straightforward fabrication of convex, straight, and concave nanopore shapes that can be employed in various technologies and applications. We explored their performance as ionic nanochannels and investigated the dependence of the ionic current rectification on the nanopore geometry. We found hysteresis in the ionic conductance that enables potential applications of the nanopores in memristors. We also investigated the dielectric oxide nanopores for DNA sensing by measuring both cis-trans and trans-cis translocations and support our data with numerical simulations based on the Poisson-Nernst-Planck model.