This dissertation presents a search for new heavy resonances decaying to a pair of Higgs bosons in proton-proton collisions at a center-of-mass energy of 13 TeV. The data were collected with the CMS detector at the LHC in 2016--2018, corresponding to an integrated luminosity of 138 $\mathrm{fb}^{-1}$. Resonances with a mass between 0.8 and 4.5 TeV are considered using events in which one Higgs boson decays into a bottom quark pair and the other into final states with either one or two charged leptons. Specifically, the single-lepton decay channel $\mathrm{HH} \to \mathrm{b}\bar{\mathrm{b}} \mathrm{WW}^* \to \mathrm{b}\bar{\mathrm{b}} \ell \nu \mathrm{q}\bar{\mathrm{q}}$ and the dilepton decay channels $\mathrm{HH} \to \mathrm{b}\bar{\mathrm{b}} \mathrm{WW}^* \to \mathrm{b}\bar{\mathrm{b}} \ell \nu \ell \nu$ and $\mathrm{HH} \to \mathrm{b}\bar{\mathrm{b}} \tau\tau \to \mathrm{b}\bar{\mathrm{b}} \ell \nu \nu \ell \nu \nu$ are examined. The signal is extracted using a two-dimensional maximum likelihood fit of the $\mathrm{H}\to\mathrm{b}\bar{\mathrm{b}}$ jet mass and HH invariant mass distributions. No significant excess above the standard model expectation is observed in data. Model-independent exclusion limits are placed on the product of the cross section and branching fraction for narrow spin-0 and spin-2 massive bosons decaying to HH. The results provide the most stringent limits to date for $\mathrm{X}\to\mathrm{HH}$ signatures with final-state leptons and at some masses provide the most sensitive limits of all $\mathrm{X}\to\mathrm{HH}$ searches.