  Using the Abstract Cauchy-Kowalewski Theorem we prove that the $b$-family
equation admits, locally in time, a unique analytic solution. Moreover, if the
initial data is real analytic and it belongs to $H^s$ with $s > 3/2$, and the
momentum density $u_0 - u_{0,{xx}}$ does not change sign, we prove that the
solution stays analytic globally in time, for $b\geq 1$. Using pseudospectral
numerical methods, we study, also, the singularity formation for the $b$-family
equations with the singularity tracking method. This method allows us to follow
the process of the singularity formation in the complex plane as the
singularity approaches the real axis, estimating the rate of decay of the
Fourier spectrum.
