Optimality conditions are mathematical criteria that help determine the best solution to an optimization problem. These conditions indicate when a solution is optimal, meaning it cannot be improved without worsening another aspect. They are essential in fields like calculus, economics, and engineering to find maximum or minimum values of functions. In optimization, the most common optimality conditions include the first-order and second-order conditions. The first-order condition involves setting the derivative of a function to zero, while the second-order condition assesses the curvature of the function to confirm whether the solution is a maximum or minimum.