The latent structure found in vast, intricate, high-dimensional datasets is effectively and accurately captured by Variational autoencoders (VAEs). Existing VAE architectures struggle with issues commonly found in real-world datasets: heterogeneous data (which combines continuous and discrete variables) and incomplete data (where missing values appear randomly). This paper introduces a comprehensive framework for designing VAEs capable of accommodating incomplete heterogeneous data. Termed MI-VAE, this novel approach utilizes a key feature: likelihood models tailored for diverse data types, including real-valued, positive real-valued, interval, categorical, ordinal, and count data. Furthermore, the model is proficient at managing missing values, accurately performing both estimation and imputation functions. Moreover, when applied to supervised learning problems, MI-VAE demonstrates a competitive predictive capacity, often exceeding that of supervised models built from the same incomplete data. Furthermore, MI-VAE not only addresses missing values but also achieves competitive performance in supervised learning tasks, demonstrating its effectiveness in real-world applications.