  In this paper we study the spaces of non-compact real algebraic curves, i.e.
pairs $(P,\tau)$, where $P$ is a compact Riemann surface with a finite number
of holes and punctures and $\tau:P\to P$ is an anti-holomorphic involution. We
describe the uniformisation of non-compact real algebraic curves by Fuchsian
groups. We construct the spaces of non-compact real algebraic curves and
describe their connected components. We prove that any connected component is
homeomorphic to a quotient of a finite-dimensional real vector space by a
discrete group and determine the dimensions of these vector spaces.
