In this talk,  I will present some results on axially symmetric solutions to the Allen-Cahn equation in entire spaces.  In particular,  a  complete branch of axially symmetric entire solutions to the Allen-Cahn equation in $\mathbb{R}^{3}$ will be constructed.  The nodal sets of these solutions behave asymptotically like catenoids at the two ends of the axis. These solutions are monotone in the radial direction and even in the direction of the axis after possible translation, and have finite Morse indices. The compactness of solutions and the linearized equations are carefully investigated and play important roles in the analysis.