I studied drinking water treatment in school and worked on a treatment plant design in Ecuador, so I do not have professional experience inside the current modern industry. But as far as I know slow-sand filtration and it's variants are still commonly used. Most of the improvements come from additional processes.

Improvements on slow-sand filtration are from additional pre-treatment methods, disinfection, more advanced filtration media like activated charcoal, and additional stages of filtration using a variety of media. Some high tech treatment plants are using biological membranes for wastewater-to-drinking-water treatment, but that's quite rare as far as I know.

Here's a decent primer of the commonly used processes: https://www.cdc.gov/healthywater/drinking/public/water_treat...