The results of several searches for additional neutral and charged Higgs bosons are presented, using data collected with the CMS detector at $\sqrt{s}=13~\mathrm{TeV}$, corresponding to an integrated luminosity of $138~\mathrm{fb}^{-1}$. The searches for neutral Higgs bosons target the decays into $\tau\tau$, WW, and pairs of lighter Higgs bosons. Each of these searches observes modest excesses of data with respect to the background expectation with local significances in the range of 3.1-3.8 standard deviations. The search for a charged Higgs boson targets its decay into a W and a neutral Higgs boson. No significant excess of data with respect to the background-only hypothesis was observed.