Count data with adjusted zeros and without zero-category arise quite frequently in studies of medicine, epidemiology, public health, biology, agriculture and other disciplines such as sociology, psychology, ecology and engineering. In this talk, I will introduce a class of newly developed multivariate zero inflated / truncated / adjusted Poisson (MZIP, MZTP, MZAP) and related distributions (e.g., Type I, II, III and IV MZIP, MZTP, MZAP distributions) by the author and his postgraduates/collaborators. First, I introduce the Type I MZIP distribution to model correlated multivariate count data with extra zeros. The distributional properties and efficient statistical methods are developed. Next, I will briefly introduce Type I MZAP models (including Type I MZIP and MZTP as special cases) by accounting for both inflation and deflation at zero. Third, to address the issues of over-dispersion and under-dispersion, other developments are highlighted by replacing Poisson with negative-binomial and generalized Poisson distributions.