To describe trans-dimensional observations in sample spaces of different dimensions, we propose a probabilistic model, called the trans-dimensional random field (TRF) by explicitly mixing a collection of random fields. In the framework of stochastic approximation (SA), we develop an effective training algorithm, called augmented SA, which jointly estimates the model parameters and normalizing constants while using trans-dimensional mixture sampling to generate observations of different dimensions. Furthermore, we introduce several statistical and computational techniques to improve the convergence of the training algorithm and reduce computational cost, which together enable us to successfully train TRF models on large datasets. The new model and training algorithm are thoroughly evaluated in a number of experiments. The word morphology experiment provides a benchmark test to study the convergence of the training algorithm and to compare with other algorithms, because log-likelihoods and gradients can be exactly calculated in this experiment. For language modeling, our experiments demonstrate the superiority of the TRF approach in being computationally more efficient in computing data probabilities by avoiding local normalization and being able to flexibly integrate a richer set of features, when compared with n-gram models and neural network models.