  In this paper, a study of topological and algebraic properties of two
families of functions from the unit interval $I$ into the plane $\mathbb{R}^2$
is performed. The first family is the collection of all Peano curves, that is,
of those continuous mappings onto the unit square. The second one is the bigger
set of all space-filling curves, i.e. of those continuous functions $I \to
\mathbb{R}^2$ whose images have positive Jordan content. Emphasis is put on the
size of these families, in both topological and algebraic senses, when endowed
with natural structures.
