How does a database query optimizer work?

Query optimizer transforms logical query into efficient physical plan. Steps: parsing (syntax tree), semantic analysis (resolve names, types), logical optimization (predicate pushdown, join reordering), physical optimization (choose algorithms, indexes). Uses statistics (cardinality estimates, histograms) to estimate costs. Algorithms: dynamic programming for join ordering, branch-and-bound pruning. Generates multiple plans, estimates costs, chooses cheapest. Hints can override. Bad statistics cause poor plans.