An electronic device may have transparent members such as display cover layers and camera windows. A transparent member such as a sapphire member may be provided with an antireflection coating. The antireflection coating may have a stack of dielectric thin-film interference filter layers that form a thin-film interference filter that suppresses visible light reflections. The stack of dielectric thin-film interference filter layers may have thicknesses and materials that provide the thin-film interference filter and coating with low light reflection properties while enhancing scratch resistance. An adhesion layer may be used to help adhere the stack of thin-film interference filter layer to the transparent member. An antismudge coating such as a fluoropolymer coating may be used to reduce smudging. Graded layers and layers with elevated hardness values may be used in the coating.