As Voyager comes under attack by unknown vessels, equipment and weapons begin disappearing from the ship. The enemy is using a high-energy version of the transporter beam to locate items of technological value and remove them. Captain Janeway and the crew track their stolen goods to an alien world that appears to be an active center of commerce. As Tuvok and Janeway beam down to search, the Vulcan immediately locates an item with a Starfleet signature: it's Leonardo da Vinci, from Janeway's Florence (not Venice) holodeck program. Somehow, the crooks got his program when they took the ship's computer processor, and the program was downloaded into the Doctor's stolen mobile emitter. He briefly shows them around the alien town. His program is designed to only think in 16th century terms so he is astounded at the technology and quickly has curried favour with one of the local trade leaders.
Da Vinci leads them to a room filled with other stolen goods and speaks of his "patron," a "prince" who provides him with everything he needs. Back on Voyager, Chakotay interrogates one of the traders who was tricked into coming aboard while looking for a warp assembly. Chakotay learns that a man named Tau sells weapons and technology he confiscates from passing ships. As it turns out, Tau is Leonardo Da Vinci's patron and Janeway gets the inventor to bring her to one of Tau's parties. She poses as a buyer and asks about computers. Tau reveals that he has Voyager's missing computer for sale, but the price would be a warship.
Armed with the very accurate topographic maps that Leonardo made of the region, Tuvok returns to Voyager and reviews the information. He and Seven of Nine locate the storage facility where the processor is kept, but a dispersion field around it makes transport impossible. Janeway will have to get inside the facility and initiate a power surge that will produce a signal strong enough for the transporter beam to lock on to. Unfortunately, Tau overhears Janeway talking to the ship and trains a weapon on her. Da Vinci comes to her defense and knocks out Tau, then he and Janeway head for the facility.
Leonardo is still fascinated by the city (which he believes initially to be America) and comments on the plumbing, saying "This is how cities should be built, using nature as a guide" in reference to the veins and arteries of a body being similar in purpose.
Once they find the processor, Janeway follows through on Tuvok's plan of creating a power surge to nullify/disable the dispersion field. The arrival of an armed guard prevents the two of them from beaming up along with the computer. Leonardo is shot, but undamaged physically, not understanding how he survived being shot. Janeway knocks out the guard, and eventually explains to him that it's a poor student that doesn't eventually surpass their master.
Janeway and da Vinci are then forced to take escape into their own hands, once they figure out that more armed guards would be on the way. Instead of teleporting back to the ship, they use a site-to-site transporter, and beam themselves a good distance away into the countryside. Still fleeing from Tau and one of his guards they board a fixed wing glider constructed by da Vinci (partially inspired by Janeway) and take off just as Tau's men begin shooting at them. Finally, Voyager is able to get close enough to the planet to beam aboard the Captain and her mentor, along with the glider that saved their lives.