We consider the possible observation of Fast Radio Bursts (FRBs) with planned future radio telescopes, and investigate how well the dispersions and redshifts of these signals might constrain cosmological parameters. We construct mock catalogues of FRB dispersion measure (DM) data and employ Markov Chain Monte Carlo (MCMC) analysis, with which we forecast and compare with existing constraints in the flat $Lambda$CDM model, as well as some popular extensions that include dark energy equation of state and curvature parameters. We find that the scatter in DM observations caused by inhomogeneities in the intergalactic medium (IGM) poses a big challenge to the utility of FRBs as a cosmic probe. Only in the most optimistic case, with a high number of events and low IGM variance, do FRBs aid in improving current constraints. In particular, when FRBs are combined with CMB+BAO+SNe+$H_0$ data, we find the biggest improvement comes in the $Omega_{mathrm b}h^2$ constraint. Also, we find that the dark energy equation of state is poorly constrained, while the constraint on the curvature parameter $Omega_k$, shows some improvement when combined with current constraints. When FRBs are combined with future BAO data from 21cm Intensity Mapping (IM), we find little improvement over the constraints from BAOs alone. However, the inclusion of FRBs introduces an additional parameter constraint, $Omega_{mathrm b}h^2$, which turns out to be comparable to existing constraints. This suggest that FRBs provide valuable information about the cosmological baryon density in the intermediate redshift Universe, independent of high redshift CMB data.
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