Let d≥2 be a positive integer, K an algebraically closed field of characteristic not dividing d, n≥d+1 a positive integer prime to d, f(x)∈K[x] a degree n monic polynomial without repeated roots, Cf,d:yd=f(x) the corresponding smooth plane affine curve over K, and Cf,d a smooth projective model of Cf,d. Let J(Cf,d) be the Jacobian of Cf,d. We identify Cf,d with the image of its canonical embedding into J(Cf,d) (such that the infinite point of Cf,d goes to the zero of the group law on J(Cf,d)). Earlier the second named author proved that if d=2 and n=2g+1≥5, then the genus g hyperelliptic curve Cf,2 contains no torsion points of orders lying between 3 and n-1=2g. In the present paper we generalize this result to the case of arbitrary d. Namely, we prove that if P is a torsion point of order m>1 on Cf,d, then either m=d or m≥n. We also describe all curves Cf,d having a torsion point of order n.