So, my offer just got accepted. Should I start packing, call the movers and give my landlord a notice?

Are we there yet? NO. Not quite.
It's fantastic that your offer is accepted. Congratulations. But  that's just the beginning of a journey which might not end exactly the way you planned it. 

Generally speaking, there can be three different scenarios after your offer gets accepted (we call it acceptance):
-Smooth Sailing Scenario and getting the key to your home just as you wished. Good for you. (Obviously you have used Bahman as your agent;))
-Bumpy Road Scenario and still getting the key to your home just as you had wished. Maybe escrow closes later than your expected date. But it does close and you get the key. Congratulations! 
-Not Closing Scenario. Yes, this is also a possible scenario. Remember that there is a reason for escrow period being 30-45 days. Many things must happen for "closing" to happen. 

"So, what things might happen after acceptance that might keep me from getting the key to my future home?" I'm glad you asked. 
These are some of the common things that might cause a "bump on the road" or a "total road blockage":
-You, the buyer, might do an inspection on the house and find out that there are things wrong with the house which you weren't aware of  and which seller doesn't want to fix  or give you credit for them  or reduce the price for. Example: Double pane windows had condensation in them and cost $10,000 to fix. Or lots of termite and dry rot damage, old and broken fence post, plumbing, roof, etc. 
-It's possible that the house does not appraise: You offered $400,000 on the property and seller accepted it  but when appraiser looked at it he only appraised it for $380,000. Now what? You can ask the seller to reduce the price to $380,000 and maybe they don't and you can't or don't want to come up with $20,000 in cash to make it happen. 
-Lender might decide that they don't want to lend to you. Even though they pre-qualified you then, after a careful review of your finances, they decided that they don't want to lend to you. Oh ya. That is possible. We see it all the time. That's why I want my buyers to work with my lender, a good lender, vs one who might not be a "good lender". And yes, they are out there. Trust me on that. 

Acceptance is great but please hope for the best and be prepared for the worst. I always tell my clients that it is not over until its recorded and their name is on the deed of the house.