  We construct a family of self-similar Markov martingales with given marginal
distributions. This construction uses the self-similarity and Markov property
of a reference process to produce a family of Markov processes that possess the
same marginal distributions as the original process. The resulting processes
are also self-similar with the same exponent as the original process. They can
be chosen to be martingales under certain conditions. In this paper, we present
two approaches to this construction, the transition-randomising approach and
the time-change approach. We then compute the infinitesimal generators and
obtain some path properties of the resulting processes. We also give some
examples, including continuous Gaussian martingales as a generalization of
Brownian motion, martingales of the squared Bessel process, stable L\'{e}vy
processes as well as an example of an artificial process having the marginals
of $t^{\kappa}V$ for some symmetric random variable $V$. At the end, we see how
we can mimic certain Brownian martingales which are non-Markovian.
