I present lattice-QCD calculations from the Fermilab Lattice and MILC Collaborations of the hadronic form factors for the semileptonic decays D→πℓν, D→Kℓν, and Ds→Kℓν. The calculation uses the highly improved staggered quark (HISQ) action for all valence and sea quarks and includes Nf=2+1+1 MILC ensembles with lattice spacings ranging from a≈0.12 fm down to 0.042 fm. At most lattice spacings, an ensemble with physical-mass light quarks is included. The HISQ action allows all the quarks to be treated with the same relativistic light-quark action, allowing for nonperturbative renormalization using partial conservation of the vector current. Results for CKM matrix elements are provided, with QCD uncertainties comparable to experimental-measurement uncertainties. An outlook to extending this work to B mesons is included.