How do formula fields work in Salesforce? What are cross-object formulas and their limitations?

Formula fields calculate a value dynamically at runtime using field values from the same record or related records. They are read-only and are not stored in the database.
Cross-object formulas: Reference fields on a related object via the relationship field name. Syntax: Account.BillingCity or via a lookup: Lookup_Field__r.FieldName__c. Cross-object formulas work up to 10 relationship levels deep.
Limitations:
— Cannot reference fields that could create circular dependencies (e.g., a formula that references another formula on the same object which references back).
— Cannot reference fields in child relationships (only parent via lookup/master-detail).
— Null handling: use ISBLANK() for text fields (ISNULL works on numbers only). Unchecked nulls can cause formula errors (e.g., division by zero).
— Maximum formula length: 3,900 characters (compiled), 5,000 characters uncompiled.
— Formula field types: Text, Number, Currency, Percent, Date, Date/Time, Checkbox.