The 1D De Gregorio model on the real line is a simplified model for the 3D incompressible Euler equations, proposed to study the competition between advection and vortex stretching. In this talk, we present some recent results on self-similar finite-time blowup solutions of this model. In an earlier work, we proved the existence of a nonlinearly stable self-similar profile that leads to a finite-time blowup from smooth initial data with finite energy, using a hybrid strategy of analysis and computer-aided proof. Very recently, we showed that there actually exist infinitely many compactly supported, self-similar solutions that are distinct under rescaling and will blow up in finite time. These self-similar solutions fall into two classes: the basic class and the general class. The basic class consists of countably infinite solutions that are eigenfunctions of a self-adjoint compact operator. In particular, the leading eigenfunction coincides with the nonlinearly stable self-similar solution previously obtained by numerical approaches. The general class consists of more complicated solutions that can be obtained by solving nonlinear eigenvalue problems associated with the same compact operator.