More recently Machine learning has really enhanced what you can do with regression. For example multivariate regressions when there are non-linear (or partially linear) relationships between feature and target variables.
For example recent regression problem involved a chemical reaction. It was suspected that a particular feature above a threshold began to display non linear behavior but it was difficult to pinpoint exactly where it began departing from linearity. ML was very helpful analyzing this.
Other than regressions and timeseries forecasting I think it's worth knowing about K-means clustering and PCA (Principal Component Analysis)/ PLS (Projection to latent structures) as well.
I've found PCA to be pretty unknown but very useful I've had success using it in the past and found it useful to explain the relationship not just between the data features and the target variable but also how the features relate to each other.