The solution of inverse problems in cardiovascular mathematics is computationally expensive. In this paper we apply a domain parametrization technique to reduce both the geometrical and computational complexity of the forward problem, and replace the finite element solution of the incompressible Navier-Stokes equations by a computationally less expensive reduced basis approximation. This greatly reduces the cost of simulating the forward problem. We then consider the solution of inverse problems in both the deterministic sense, by solving a least-squares problem, and in the statistical sense, by using a Bayesian framework for quantifying uncertainty. Two inverse problems arising in haemodynamics modelling are considered: (i) a simplified fluid-structure interaction model problem in a portion of a stenosed artery for quantifying the risk of atherosclerosis by identifying the material parameters of the arterial wall based on pressure measurements; (ii) a simplified femoral bypass graft model for robust shape design under uncertain residual flow in the main arterial branch identified from pressure measurements.