Given a measure on the real line, there are many polynomial families naturally associated to it. From the combinatorial point of view, some interesting properties of these families are orthogonality (and more general diagram formulas), generating functions, recursion relations etc. I will describe a new such family, of free Appell polynomials, for which all of the above have very nice explicit formulas. In fact, one can associate a free Appell family to any functional on a (non-commutative) algebra. The resulting polynomials can be considered as analogs in Voiculescu's Free Probability Theory of the more familiar Appell polynomials. No probability background will be assumed.