Superhumans have existed throughout history, usually being seen as mythological beings.  Gods, angels, demons, or other such things were, in the view of modern historians, just superhumans which the people of the past didn't understand.  They believe that the reason pantheons of deities tended to be family trees is that superhumans would work together to rule over past societies, but their decedents wouldn't always get powers, leading to the pantheons losing their power within a few hundred years.

Modern gene mapping revealed three separate "Catalyst" genes, called that because they trigger alterations in the human body.  The Energetic variation alters the metabolism of a person's body, allowing them to utilize, store, or control different forms of energy.  The Mental Variation effects the brain and sometimes the nervous system, altering perceptions and granting psionic abilities.  The Physical Variation can cause changes to most of the human body, altering appearance, giving people enhanced muscles, and the like.  It can cause the most varied changes of the three.  A person can have any combination of the three or all three, and multiple copies of the same gene will only make them stronger.  A parent with an active gene has a 50% chance of passing it to their child, and often it becomes inactive when passed.

While best estimates of past human genetics show that the rate of Catalyst genes within the human population hasn't increased drastically in the last century, the appearance of large numbers of superhumans began in the 1930s.  Scientists believe that the activation of the gene requires high amounts of stress or a near-death experience, and that the gene responds by adapting the person in a way that negates some of all of the stress.   A person that is trapped in a burning building, for example, may adapt to be able to breath in reduced oxygen environments, be less effected by smoke, become resistant to heat, or even develop the ability to manipulate the flames depending on their version of the gene.  These events activated the catalyst genes far less frequently before the 1930s, though, which baffles scientists studying the phenomenon.

Roughly one in one hundred people have a Catalyst gene, roughly one in ten thousand have two different catalyst genes, and roughly one in a million have all three.  Only about one percent of people with a gene have an active form, however, meaning that there are roughly 80 million superhumans on Earth.  They are common enough for everyone to have met at least one, even in small towns, though rare enough that their skills are in demand.

The existence of superhumans has accelerated the development of technology as well.  Energy weapons, advanced materials, force fields, and advanced spacecraft exist, though humans haven't figured out how to travel faster than light yet.

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Possible plots for this world: (Suggest something else if you have another idea that could work)

Alien invasion
We are arrested for crimes, false or real, and sent to a prison colony.  We band together to survive the other inmates.
Abducted by aliens, sent to hostile planet, need to survive and defeat them to get back.
We are hired for some kind of job (depends on the characters).