Markov Chain Monte Carlo (MCMC) methods often have difficulties in moving between isolated modes. To understand these difficulties, some MCMC theory and some mode jumping approaches will be reviewed, first in fixed dimension and later in variable dimension. The focus will lie on improving the eficiency of the powerful, but computationally expensive method "tempered transitions". A technique for optimising the method's parameters ("temperatures") will be proposed. It will be demonstrated that the default choice of geometric temperatures can be far from optimal. The tuning technique will then be tested on a hard applied sampling problem, namely on sampling from a fixed-dimensional mixture model. The results will show that the optimisation is robust and performs well and that tempered transitions achieves mode jumping ("label-switching") where standard MCMC fails. Since mixture models are often of variable dimension, it will be verified that tempered transitions and the tuning technique can also be applied in variable-dimensional problems. Tests on a variable-dimensional mixture model will confirm that tempered transitions also improves jumps between dimensions.