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We constrain the rest-frame FUV (1546Å), NUV (2345Å) and U-band (3690Å) luminosity functions (LFs) and luminosity densities (LDs) with unprecedented precision from $z\sim0.2$ to $z\sim3$ (FUV, NUV) and $z\sim2$ (U-band). Our sample of over 4.3 million galaxies, selected from the CFHT Large Area $U$-band Deep Survey (CLAUDS) and HyperSuprime-Cam Subaru Strategic Program (HSC-SSP) data lets us probe the very faint regime (down to $M_\mathrm{FUV},M_\mathrm{NUV},M_\mathrm{U} \simeq -15$ at low redshift) while simultaneously detecting very rare galaxies at the bright end down to comoving densities $<10^{-5}$ Mpc$^{-3}$. Our FUV and NUV LFs are well fitted by single Schechter functions, with faint-end slopes that are very stable up to $z\sim2$. We confirm, but self-consistently and with much better precision than previous studies, that the LDs at all three wavelengths increase rapidly with lookback time to $z\sim1$, and then much more slowly at $1<z<2$--$3$. Evolution of the FUV and NUV LFs and LDs at $z<1$ is driven almost entirely by the fading of the characteristic magnitude, $M^\star_{UV}$, while at $z>1$ it is due to the evolution of both $M^\star_{UV}$ and the characteristic number density $ϕ^\star_{UV}$. In contrast, the U-band LF has an excess of faint galaxies and is fitted with a double-Schechter form; $M^\star_\mathrm{U}$, both $ϕ^\star_\mathrm{U}$ components, and the bright-end slope evolve throughout $0.2<z<2$, while the faint-end slope is constant over at least the measurable $0.05<z<0.6$. We present tables of our Schechter parameters and LD measurements that can be used for testing theoretical galaxy evolution models and forecasting future observations.