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At Medfield College, science buff Dexter Riley and his friends, including Richard Schuyler and Debbie Dawson, eavesdrop via a hidden walkie-talkie on a board meeting led by Dean Eugene Higgins, discussing the small college's continuing precarious finances. Later that afternoon, Professor Lufkin shows Higgins around the science lab where Dexter is working on an experiment with invisibility and another student, Druffle, explores the patterns of the lives of bumblebees. That night, unknown to anyone, during a powerful thunderstorm, the roof of the lab is struck by lightning, sending a current of electricity down a metal beam and through Dexter's complex experiment components. The next day, as Dexter examines his burnt equipment with dismay, Higgins meets with businessman A. J. Arno who has recently purchased Medfield's mortgage. When Dexter accidentally drops one half of his protective glasses into a container of his experimental formula, it appears as if the substance destroys them, but upon closer examination, Dexter realizes the frames are merely partially invisible. After several excited tests, Dexter boldly places his fingers in the liquid and they disappear. Schuyler and Debbie arrive and are horrified to see Dexter with a partial hand, but Dexter insists Schuyler test the substance as well, admitting only afterward that he does not yet have an antidote. Just then, Higgins brings Arno to visit the lab, stunning the students, as only two years earlier, Dexter was instrumental in exposing Arno's crooked gambling scheme. Although distracted by the condition of their partially visible hands, Dexter and the others notice that Arno is more concerned with the campus architecture than Higgins' curriculum speech. After Arno and Higgins depart, Dexter and Schuyler discover, to their surprise, that water washes off the invisibility formula's effects. Curious about Arno's behavior, Dexter convinces Schuyler to use the invisibility formula to sneak into Arno's office that night to look around. Although they are nearly discovered when Schuyler steps into a puddle, making his tennis shoes visible, the boys get inside Arno's office where they find a model of Medfield College redesigned as sprawling gambling establishment. After taking photos of the model, the boys flee with Debbie's help. The next day Dexter shows the photos to Lufkin and Higgins, both of whom are alarmed and concerned about how to buy back the college's mortgage. Convinced that Druffle's bumblebee study would draw attention and investments to Medfield, Higgins reacts angrily when Dexter assures him that his invisibility formula could win the top prize money in the upcoming Forsythe science contest. Not having admitted to anyone that Medfield has been dropped from the contest for being too insignificant, Higgins determinedly contacts the contest's sponsor, millionaire Timothy Forsythe and agrees to meet over a game of golf, despite his inability to play. Upon learning of Higgins' plan and suspecting it must be connected with raising money for the college, Dexter urges Schuyler to volunteer to serve as Higgins' caddy while, hidden by the invisibility formula, he will take control of Higgins' golf ball. At the golf club, Forsythe and the state university dean, Collingsgood, are amazed by Higgins' quirky golfing abilities, which include numerous hole-in-one-shots, as is Arno who is also at the club. After the game, Forsythe enthusiastically agrees to reinstate Medfield into the competition for the Forsythe Award. Meanwhile, Arno accidentally sees Dexter becoming visible in the club showers and grows suspicious. When the local television news covers Higgins' extraordinary golf game, he is invited to join an exclusive tournament in nearby Ocean City. Convinced that he will win enough money to pay the college's mortgage, Higgins brashly accepts and that afternoon departs with Schuyler. Learning of the tournament from Druffle too late, Dexter misses the plane and is forced to watch the competition on television where Higgins' game against two professionals is a disaster. In his business office, Arno and his henchmen, Cookie and Alfred, also watch the tournament and ponder Higgins' odd inconsistency. Upon returning to the college, Higgins tells Lufkin that Druffle's bee experiment is the school's last chance. Both men are stunned when just then Druffle appears swathed in bandages after being attacked by the bees, to which he is allergic. Hoping to assuage the crestfallen Higgins, Lufkin suggests that they give Dexter's unproven formula a chance and the dean reluctantly agrees. That evening, Cookie, disguised as a janitor, sneaks into the campus lab where he witnesses Dexter and Schuyler using the invisibility spray, and reports to Arno, who orders him to return and steal it. The following day, Forsythe and members of his committee arrive on campus to judge the best science experiment at the college. Unaware that their spray bottle has been replaced by Cookie, Dexter and Schuyler make their presentation and are stunned when it has no effect. Disappointed and angry, Forsythe and Higgins depart as Dexter remains confused until he chats with Charlie, the janitor. Learning that there is no night janitor, Dexter realizes that Cookie was a spy and likely stole the formula. Concluding that Arno must be behind the theft, Dexter plants a walkie-talkie in his office. A couple of days later, Schuyler overhears Arno plotting with Cookie to rob Medfield bank by making themselves and the money invisible. Certain that if he could retrieve the formula before the Forsythe Award announcement that night, he could still win the contest, Dexter sends Schuyler to the police and goes to inform the bank's president, Wilfred Sampson. When both the police and Sampson dismiss the boys' story about invisibility, Dexter and his friends gather outside of the bank to make plans. While an invisible Arno and Cookie knock out the guards and take the money, Dexter unsuccessfully tries to use a fire hydrant to hose the men down as they exit the bank. When Sampson realizes a theft has occurred, he contacts the police who join the college students in a wild chase of the car driven by the invisible robbers. After briefly eluding everyone, Arno orders Cookie to make the car invisible, but they are spotted on a dirt road in a park. Deducing Arno will not leave town but go to his home instead, Dexter drives there and forces Arno's car into a neighboring pool where it, the money and the men become visible. Taking the formula, Dexter and the others dash to the presentation of the Forsythe Award and plead for one more opportunity to demonstrate their invention. Frustrated by Dexter's determination, Higgins intervenes just as Dexter sprays Schuyler, and, again, there is no result. Realizing the dip in the pool has diluted the formula, Dexter tries to explain to Forsythe just as the top half of Higgins becomes invisible, thus shocking the group and winning the top prize to save Medfield for another year.