This talk classifies all harmonic noncommutative polynomials, as well as all polynomial solutions to other selected noncommutative partial differential equations. The directional derivative of a noncommutative polynomial in the direction $h$ is defined as $D[p(x_1,\ldots,x_g),x_i,h]:= \frac{d}{dt}[p(x_1, \ldots, (x_i+th), \ldots, x_g)]_{|_{t=0}}$. From this noncommutative derivative, one may define differential equations which take as solutions polynomials in free variables. A noncommutative harmonic polynomial is a polynomial such that its noncommutative Laplacian is zero.