Quest: Seniors

Seniors: Unlocked at Level 23 to unlock the ability to age Adult Sims into Senior Sims.

This quest-line is time-limited, if you finish within the given time of 4 days you'll receive the reward. 
Reward: Exercise Cycle. (Available in Online Store, if you do not finish the quest on time.)
If you do not finish the quest on time, you still have access to making Seniors.

Quest: Ghost Hunters

Ghost Hunters: Unlocked at Level 22 to unlock the Ghost Hunter hobby and make Haunted items available for purchase in the Home Store.
 

This is a time-limited quest-line, you're given 4 days.
Reward: The "Haunted House" housing lot. (Available in Online Store if you did not finish in time).

Quest: Adulthood

Adulthood: Unlocked at Level 21
This quest allows you to unlock the ability to age Teen Sims into Adult Sims. 

This is a time-limited quest-line, you're given 6 days.
Reward: Fashion Pack (makes some Teen-only clothing options available to Adult Sims). (Available in Online Store if you did not finish in time).

Quest: Higher Education

Higher Education: Unlocked at Level 19 to gain access to the High School building on the Town Map.

No time limit or reward for this quest.

Quest: Teenagers (Coming of Age)

Coming of Age (Teenagers): Unlocked at Level 18

This (no-time-limit) quest unlocks the ability to age Preteen Sims into Teens.

There is no time-limit for this quest and no reward!

Quest: Multi-Story Renovations

Multi-Story Renovations: Unlocked at Level 17 to unlock the ability to add additional stories to your Sims' Houses. (This quest is a requirement for the DIY Balconies discovery quests!).
 

This is a time-limited quest-line, you're given 6 days (5 days 23 hours remaining at the start).
Reward: Elevators & Escalators pack. (Available in Online Store if you did not finish in time).

Quest: An Alien Concept: Weather Machines

An Alien Concept: Weather Machines: Unlocked at Level 14 to gain access to the Weather Machine and Snow Park.

This is a time-limited quest-line, you're given 2 days.
Reward: Winter Clothing pack for the Costume & Swim Store on the Town Map and Winter Accessories for your Sims (accessible when you create a Sim or through a Wardrobe). (Available in Online Store if you did not finish in time).

Quest: Preparing for Preteens

Preparing for Preteens: Unlocked at level 13 to unlock the ability to age Toddler Sims into Preteens.

There is no time-limit for this quest and no reward!

Quest: It's all going Swimmingly

It's All Going Swimmingly: Unlocked at level 13

This is a time-limited quest-line, you're given 2 days to unlock the ability to build Pools in your house lots.
Reward: Volleyball Net for pools (Available in Online Store if you did not finish in time).
This quest was last verified with the Family Furnishings update (July 2019).

Quest: A Quest for Toddlers

A Quest for Toddlers: Unlocked at level 12

This is a time-limited quest-line, you're given 2 days to unlock the ability to grow your babies into Toddlers, as well as have access to the "Adopt a Toddler" purchase action available via Sim's phones within the game.
Reward: The Dress-Up Chest (Available in Online Store if you did not finish in time).
This quest was last verified on the Family Furnishings update (July 2019).

Quest: Raiders of the Lost Artifacts

Raiders of the Lost Artifacts: Unlocked at level 10 to unlock Monuments and Resources from the Mystery Island. [I know it's spelled wrong, but that's EA/FM's fault.]

This is a time-limited quest-line, you're given 2 day (23 hours remaining at the start). 
Additional reward: Ability to collect the Treasure Chest prize pack (not available in Online Store).

Quest: The Mysterious Island

The Mysterious Island: Unlocked at Level 10

This is a time-limited quest-line, you're given 2 days. 
Reward: Island Replica Sentinel (Golem) is unlocked for purchase in the Home Store (not available in Online Store).

Quest: Ocean View Estate

Ocean View Estate: Unlocked at Level 9

This is a time-limited quest-line, you're given 2 days (1 day 23 hours remaining at the start) to unlock the Beachside Escape Premium House.
If you do not finish the quest on time, you still have access to the beach side lots. The house will be available in the Online Store.

This quest was last verified on the Glitz and Glam update (Sept. 2015).

Quest: Two and a Half Sims

Two and a half Sims: Unlocked at level 8

This quest-line is time-limited, if you finish within the given time of 3 days (2 days and 23 hours remaining at the start) you'll receive the reward. In this case get access to the items.

Since the Pregnancy update it's not longer neccessary to have a married couple to add a baby.

This quest has been last verified on the Family Furnishings update (July 2019)
Reward: Limited-edition Baby Decoration Pack (not available in Online Store)
If you do not finish the quest on time, you still have access to making babies.

Quest: Money Grows on Trees?

Money Grows on Trees: Unlocked at Level 7

This quest-line is not time-limited and unlocks the Simoleon Sprout and (access to) the LP Lotus (which costs Real Money!)

This quest has been last verified on the Family Furnishings update (July 2019)

Quest: Love is in the Air

This quest is available from level 6 onwards.

This used to be the very first quest line in the game, now it's the second. You're given 2 days (shown as 1 day and 23 hours remaining) to complete this quest and receive the Wedding Bundle. 
If you do not manage to complete this quest in time, you still have access to marriage.

Hobbies: Bird Feeding

Note: If you have not done the Bird Feeding quest (available here), you don't have access to this hobby.
Only Senior Sims can do this hobby!

Quest: Bird Feeding

Bird Feeding Quest: (Unlocked at Level 24)

Bird Feeding/Befriending:
Your Senior Sims are in the mood for some bird feeding. Complete this quest and unlock a huge range of friendly, sometimes exotic birds for your Senior Sims to discover!
(Complete this quest within the time limit and unlock your very own UNCAGED BIRD! Teach it tricks or have a chat! No one will judge.)

[Note: The first description talks about a range of birds, therefore the hobby could have been called "Bird Spotting" as you need to find certain birds. Internally (game files) it's called "Pigeon Befriending". There is only one bird (Parrot on a Perch) available if the quest is completed within the time limit.]

Quilting Quest: In Stitches

Quilting Quest (In Stitches): Unlocked at Level 25 
This quest unlocks the Quilting hobby and the time-limited Patchwork Teddy.

Complete this set within the time limit 4 days to unlock an exclusive patchwork teddy.