  We study the N=1 version of Argyres-Douglas (AD) points by making use of the
recent developments in understanding the dynamics of the chiral sector of N=1
gauge theories. We shall consider N=1 U(N) gauge theories with an adjoint
matter and look for the tree-level superpotential W(x) which reproduces the N=2
AD points via the factorization equation relating the N=1 and N=2 curves. We
find that the following superpotentials generate the N=2 AD points: (1)
W'(x)=x^N \pm 2\Lambda^N, (2) W'(x)=x^n, N-1\ge n \ge N/2+1.
  In case (1) the physics is essentially the same as the N=2 theory even in the
presence of the superpotential. There seems to be an underlying structure of
N-reduced KP hierarchy in the system.
  Case (2) occurs at the intersection of a number of N=1 vacua with massless
monopoles. This branch of vacua is characterized by having s_+=0 or s_-=0 where
s_{\pm} denotes the number of double roots in P_N(x)\pm 2\Lambda^N. It is
possible to show that the mass gap in fact vanishes at this AD point. We
conjecture that it represents a new class of N=1 superconformal field theory.
