Some variations provide a method of making an additively manufactured single-crystal metallic component, comprising: providing a feedstock comprising a first metal or metal alloy; providing a build plate comprising a single crystal of a second metal or metal alloy; exposing the feedstock to an energy source for melting the feedstock, generating a melt layer on the build plate; and solidifying the melt layer, generating a solid layer (on the build plate) of a metal component. The solid layer is also a single crystal of the first metal or metal alloy. The method may be repeated many times to build the part. Some variations provide a single-crystal metallic component comprising a plurality of solid layers in an additive-manufacturing build direction, wherein the plurality of solid layers forms a single crystal of a metal or metal alloy with a continuous crystallographic texture. The crystal orientation may vary along the additive-manufacturing build direction.