The lognormal distribution with parameters µ and σ²may be defined as the distribution whose logarithm is normally distributed with mean µ and variance σ².The lognormal distribution is positively skewed and heavy-tailed. It has many applications not only in the natural sciences but also in social science, engineering, medical science and pharmacy. The theory of lognormal distributions and its applications in environmental sciences are studied. As a particular application, the estimation of mean rain rate by the threshold method was studied in the Tropical Rainfall Measuring Mission Project, a United States-Japan joint venture. Data sets are supposed to be obtained with space-borne precipitation radar. Others are: level crossing methods in time series, compound distributions in actuarial sciences and parameter estimation with missing observations in bivariate normal distributions. Directional statistics is also studied. Typical examples are the statistical analysis of wind direction, wind speed and wave height in environmental sciences and that of the location of bacterial genes in molecular biology. The study includes the theory of statistical distributions on the unit sphere, the cylinder, the torus and the complex unit sphere.