We study some mathematical properties of a new generator of continuous distributions with one extra parameter called the odd power Cauchy family including asymptotics, linear representation, moments, quantile and generating functions, entropies, order statistics and extreme values. We introduce two bivariate extensions of the new family. The maximum likelihood method is discussed to estimate the model parameters by means of a Monte Carlo simulation study. We define a new log-odd power Cauchy-Weibull regression model. The usefulness of the proposed models is proved empirically by means of three real data sets.