We show that $(\textbf{E},\textbf{H})=({\textbf{E}_0},{\textbf{H}_0}){e^{i[{k_0}S(\textbf{r})-\omega t]}}$(E,H)=(E0,H0)ei[k0S(r)-ωt] is an exact solution to the Maxwell equations in free space if and only if $\{{\textbf{E}_0},{\textbf{H}_0},\nabla S\}${E0,H0,∇S} form a mutually perpendicular, right-handed set and $S(\textbf{r})$S(r) is a solution to both the eikonal and Laplace equations. By using a family of solutions to both the eikonal and Laplace equations and the superposition principle, we define new solutions to the Maxwell equations. We show that the vector Durnin beams are particular examples of this type of construction. We introduce the vector Durnin-Whitney beams characterized by locally stable caustics, fold and cusp ridge types. These vector fields are a natural generalization of the vector Bessel beams. Furthermore, the scalar Durnin-Whitney-Gauss beams and their associated caustics are also obtained. We find that the caustics qualitatively describe, except for the zero-order vector Bessel beam, the corresponding maxima of the intensity patterns.