If you are paying rent for your living quarters, the answer is yes, at least the great majority of the time.
Renter’s insurance covers you for your personal property (computers, furniture and other personal belongings) in the event of a covered loss or theft. So if your apartment burns down and you lose everything, your renter’s insurance policy would cover the replacement of that property.
If you don’t have the policy, you will simply be out of luck, since your landlord’s insurance will specifically not cover your property as the renter.
Renter’s policies generally will cover your property even if it is not located on the rental premises. That may apply if you have a mini-storage unit you are using or some of your property is stolen/damaged when it is out and about with you.
Renter’s policies also provide liability protection to help you in the event you get sued due to your negligence. There are a million ways this could apply, but one specific example is you not cleaning up your dirty laundry on the floor, which causes a guest to trip, break his leg and sue you for the injury and lost wages since he also can’t work anymore.
Renter’s insurance is relatively very cheap, only $10-30 a month. You may even save money on your overall insurance expenses by bundling the renter’s insurance with your auto insurance policy because of the multi-line discounts you can get by using the same company.