What should I do if my baby will not sleep in a crib?

Check crib. Make sure that crib is comfortable, that the room is not too cold or too warm, no scary things around. The next steps will take a few days but you need to do it. Put the baby in the crib. Leave the room. Let him/her cry a little bit. You can return to the room, try not to pick the baby up. No yelling, no screaming, no body punishment, just make it clear that the baby has to sleep in the crib.
Depends. Adjustment to a crib with crying is normal; a child of any age can safely cry unattended in crib for 5-10 minutes and may self-comfort. Children older than 8-9 months may need to be left for longer. For older infants, a transitional object like teddy bear may help (avoid toys with small infants). Discuss with pediatrician for strategies and to ensure no medical or developmental issues are at play.
No alternative. A baby will sleep and do where and when you train them to. You must first let go, and abandon the myth that you must do everything baby wants. There are some cultures where moms never leave baby, they just put them in a sling and carry them continuously. If you want to live in tribal New Guinea, fine. In the US we pace a value on individual success and that starts with learning to sleep alone.