Almost every general ophthalmologist would be trained to assess patients with astigmatism. Your doctor will comprehensively assess you to exclude any pathology, including corneal astigmatism (keratoconus, or keratoglobus, etc) or lenticular astigmatism (cataract) before giving you a suitable spectacle or contact lens prescription.
Corneal astigmatism
This refers to the irregular shape of the cornea that causes light to not be focused correctly on the retina. This results in the patient seeing unclear images. A regular, healthy cornea is perfectly round in shape.
Lenticular astigmatism
Unlike corneal astigmatism, lenticular astigmatism affects the lens instead of the cornea. Patients with lenticular astigmatism have variations in their lenses, rather than having perfect curves, that cause images to be projected onto the retina imperfectly.