We prove existence and uniqueness for two classes of martingale problems involving a nonlinear but bounded drift coefficient. In the first class, this coefficient depends on the time t, the position x and the marginal of the solution at time t. In the second, it depends on t, x and p(t,x), the density of the time marginal w.r.t. Lebesgue measure. As far as the dependence on t and x is concerned, no continuity assumption is made. The results, first proved for the identity diffusion matrix, are extended to bounded, uniformly elliptic and Lipschitz continuous matrices. As an application, we show that within each class, a particular choice of the coefficients leads to a probabilistic interpretation of generalizations of Burgers' equation.