Pandas GroupBy with a Custom Aggregation Function

Problem: Given a pandas DataFrame, group rows by a key column and apply a custom (non-built-in) aggregation function to each group. Example: groupby('category').apply(lambda g: custom_function(g)) or groupby('category')['value'].agg(custom_function). Approach: Use DataFrame.groupby(key) to form groups, then .apply() (for whole-group logic returning any shape) or .agg() (for reducing each group's column to a scalar) with the custom function passed as a callable. Discuss performance trade-offs of .apply() (row/group-wise Python callback, slower) vs vectorized alternatives.