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We report the production of quantum degenerate Bose-Bose mixtures of Cs and Yb with both attractive (Cs + $^{174}$Yb) and repulsive (Cs + $^{170}$Yb) interspecies interactions. Dual-species evaporation is performed in a bichromatic optical dipole trap that combines light at 1070 nm and 532 nm to enable control of the relative trap depths for Cs and Yb. Maintaining a trap which is shallower for Yb throughout the evaporation leads to highly efficient sympathetic cooling of Cs for both isotopic combinations at magnetic fields close to the Efimov minimum in the Cs three-body recombination rate at around 22 G. For Cs + $^{174}$Yb, we produce quantum mixtures with typical atom numbers of $N_mathrm{Yb} sim 5 times 10^4$ and $N_mathrm{Cs} sim 5 times 10^3$. We find that the attractive interspecies interaction (characterised by the scattering length $a_mathrm{CsYb} = -75,a_0$) is stabilised by the repulsive intraspecies interactions. For Cs + $^{170}$Yb, we produce quantum mixtures with typical atom numbers of $N_mathrm{Yb} sim 4 times 10^4$, and $N_mathrm{Cs} sim 1 times 10^4$. Here, the repulsive interspecies interaction ($a_mathrm{CsYb} = 96,a_0$) can overwhelm the intraspecies interactions, such that the mixture sits in a region of partial miscibility.
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