Twenty-six badges, awarded for what you actually do. Most come in five tiers.
Badges are earned, never given. Each one is worked out from your solving record every time your profile loads, so it always matches the numbers beside it — and if a figure changes, the badge changes with it.
Most badges are a family of five tiers, marked with Roman numerals: The Committed I at ten problems solved, up to V at five thousand. Your profile shows only the highest tier you have reached in each family. A few are one-off and have no tiers.
Your profile shows the badges you have earned. The full list below shows everything, and hovering or tapping a badge tells you what it takes.