Free Meixner convolution semigroups are a class of free convolution semigroups of states on the algebra of non-commutative polynomials. They also have a more explicit representation as states on operator algebras. One of the properties shared by these semigroups is that the corresponding families of orthogonal polynomials are martingales. Examples of free Meixner semigroups include certain free product states, the free multinomial distribution, and its generalizations. All of these are, or may be, the free analogs of the quadratic exponential families in statistics.