Leveling Overview
Nuclear Throne follows a simple leveling system. Radiation that drops from enemies serves as experience points which will fill the rad meter. Once filled the character will level up, and upon exiting the current level they will be given the option of one skill from a pool of four (typically) that are known as mutations.
The player will always start at level 1 at the start of the run, and every consecutive level up will grant a mutation up to level 9. This means that a typical run will have 8 mutations, but a 9th can be gained through Crown of Destiny.
Reaching level 10 is referred to as 'ultra'. This gives character-specific mutation choices, which typically gives a boost to that character's unique passive or active ability. Any additional rads gained after reaching ultra will continue to fill the rad meter and is meant to be used as additional ammunition for ultra weapons. If the rad meter is fully filled out any additional rads gained will be wasted, similar to collecting pickups at full health/ammo.
The following table lists how many rads it takes to level up, as well as the total rads collected up to that point:
Level: | Lvl. 1 | Lvl. 2 | Lvl. 3 | Lvl. 4 | Lvl. 5 | Lvl. 6 | Lvl. 7 | Lvl. 8 | Lvl. 9 | Ultra | Full Ultra |
Rads needed: | 0 Rads | 61 Rads | 120 Rads | 180 Rads | 240 Rads | 300 Rads | 360 Rads | 420 Rads | 480 Rads | 540 Rads | 600 Rads |
Total Rads: | 0 Rads | 61 Rads | 181 Rads | 361 Rads | 601 Rads | 901 Rads | 1,261 Rads | 1,681 Rads | 2,161 Rads | 2,701 Rads | 3,301 Rads |
Every level up gives a set of four mutations (five with Horror) in which the player can select one. There are 28 mutations normally along with a secret one. Once a mutation is picked it will be removed from the mutation pool for that run, which includes Patience that is a reroll mutation. The exception to this is the other reroll mutation Last Wish, as it is 'consumed' upon entering Jungle and is re-added to the mutation pool.
In a vanilla run every set can be considered random - every mutation has an equal chance of appearing. In reality this is a set mutation seed for that specific run, which matters in the context of daily runs (for players who share their sets with others), and for weeklies (which can be replayed any number of times).
Leveling Mechanics
This section will serve as a basic introduction for leveling mechanics. There is a fully in-depth guide for Daily Sheet mechanics, however some of explanations are heavily on the technical side.

The example image is from the daily sheet, but illustrates how basic leveling works in any run. The player starts from Set 1 and any time they level up, they move to the next set. (On the Betas they start from Set 0, which is relevant just for daily/weeklies).
Only the four leftmost mutations are seen upon leveling up. The fifth mutation on the right is only seen with Horror's passive. If a mutation is taken then it gets removed from the mutation pool; for example if Sharp Teeth is taken in Set 1 then it will not appear in Set 2, and Thronebutt will now appear as the 4th mutation slot instead.
If set logic seems confusing at all, a Weekly challenge can be played multiple times while taking different mutations each run while paying attention to any differences.
Taking Patience or consuming Last Wish will all advance the set number by one on Non-Beta, which is essentially leveling up. On the Betas these have a 'fixed' seeded set regardless of when the reroll mutations were taken. Any version can consider these sets as essentially random on a Normal run - only mattering for dailies/weeklies.
Non-Beta Bug: If two or more sets are seen during a level transition and Patience is taken before the last set is seen, a set is skipped and then that last set will then reappear as the Patience set upon clearing the following level. In other words the number of mutations remain the same, but a set can be selected from twice. While sets cannot be predicted in vanilla runs, if sets are known in a daily/weekly challenge it can be used to some benefit.