Non-computable functions are mathematical functions that cannot be calculated by any algorithm or computer program. This means there is no systematic way to determine their values for all possible inputs. A famous example is the Halting Problem, which shows that it is impossible to create a program that can determine whether any given program will eventually stop running or continue indefinitely. These functions highlight the limitations of computation and the boundaries of what can be solved using algorithms. While many functions can be computed efficiently, non-computable functions serve as a reminder that not all problems can be addressed through mechanical means, emphasizing the complexity of theoretical computer science.