A kernel $K: X \times Y \to \mathbb{R}$ is said to be totally nonnegative if every finite minor of the form $\det (K(x_i, y_j))_{i,j=1}^n$ is nonnegative, for any $n \geq 1$. For which functions $F: \mathbb{R} \to \mathbb{R}$ does the left composition operator $C_F(K) = F \circ K$ preserves the total nonnegativity property? We will present several characterizations of these functions. We will also discuss many natural connections between this problem, probability theory, and group representation theory.