there are many type of plants - with scrubber and slow sand filters, with pretreatment includes Al-phosphate or without them, with ozonization or UV radiation, with turbidity decreasing, with using a polymers...
yes, the processes are many. There is also the "natural" version through (constructed) wetlands. They can be compacted by chosing a good locally adapted version of microbes and macrophytes, but natural regeneration through the filtering capacity of soils and plants has also been used in multifunctional parks (water filtration and recreation). Unless space is very scarse, such solutions tend to be less costly, but require a different set of knowledges and competencies than more conventional treatment plants. In any event, it's worth analysing entire process cycles and explore whether substances which are particularly difficult/expensive to remove can be separately captured and processed. As a general rule, end of pipe solutions are much more costly than cycling (where "waste" at one step of the cycle can become a resource for another one), thus learning from nature.