  We construct a Continuous Wavelet Transform (CWT) on the torus $\mathbb T^2$
following a group-theoretical approach based on the conformal group $SO(2,2)$.
The Euclidean limit reproduces wavelets on the plane $\mathbb R^2$ with two
dilations, which can be defined through the natural tensor product
representation of usual wavelets on $\mathbb R$. Restricting ourselves to a
single dilation imposes severe conditions for the mother wavelet that can be
overcome by adding extra modular group $SL(2,\mathbb Z)$ transformations, thus
leading to the concept of \emph{modular wavelets}. We define modular-admissible
functions and prove frame conditions.
