However, there are recent developments which make possible Lorentz invariant quantization of string theory in 4-dimensional Minkowski space-time.
The obstruction to conformal symmetry is known as the Weyl anomaly and is proportional to the central charge of the worldsheet theory.
A worldsheet theory with the correct central charge can be constructed by introducing a non-trivial target space, commonly by giving an expectation value to the dilaton which varies linearly along some spacetime direction.
While clearly not of phenomenological interest, string theories in two dimensions serve as important toy models.
Such a description known as the c=1 matrix model captures the dynamics of bosonic string theory in two dimensions.
These "matrix models" are understood as describing the dynamics of open strings lying on D-branes in these theories.