At the time of my writing this answer the latest official update of Faerun seems to be dated DR1489, the last definite point in the timeline presented in Sword Coast Adventurer's Guide (SCAG). The book summarizes quite a number of world-shaking events (well, depending on what the latest you had on the world was), and if you wish to run an FR story that stays relatively close to canon (even though said canon is rather ambiguous, on purpose), it's definitely worth getting and reading through.
As for seasons and adventures, the latest we know about (again, at the time I'm writing this) is Storm King's Thunder, which has not actually been released yet, and we don't seem to have a specific in-game date for its events (yet?). Nonetheless, besides reading the official promotional material, you can also get a feel of the events it will detail (and throw at your players) by checking out Force Grey: Giant Hunters, an official(ly sponsored) actual play video series on youtube run by Matthew Mercer who uses SKT.
For previous stories, please see JAMMalcolmson's answer. :)
Master Chronology
Based on the link provided by Mark Horsfall, and the information Paul Westermeyer has compiled and compared in the linked article, the following information has been derived:
Sources
The three primary campaign settings for D&D are Greyhawk (GH), Forgotten Realms (FR), and Dragonlance (DL). These three have more material published than the other settings, and include the majority of the cross-time references. (Which other settings are: Spelljammer (SJ), Maztica (MZ, part of FR), Al-Qadim (AQ, Also part of FR), Mystara (MY), Ravenloft (RL), Planescape (PS), Birthright (BR), Athas (AT), Red Steel (RS), Odyssey (OD), Lankhmar (LM), Eberron (EB) and Conan (CO).)
There are four additional primary sources for chronological cross-time points: Spelljammer, Planescape, Ravenloft, and Dragon Magazine (DM) articles written by Ed Greenwood himself.
Please note that there are some contradictions in sources, so precedence is given to authors with more published works and to sources based on Spelljammer (as it is specifically designed to cross-bridge the settings) as well as the Big 3 (FR, GH, and DL).
FR/DL cross-time link
Based on comparative information from SJ, FR, DL, RL, and PS the best cross-time linkage is:
1361 DR <-> 358 AC
The RL data contradicts itself, weakening it as a source, and the PS data fails to trump the SJ data.
GR/FR cross-time link
Based on data from FR/RL/GH and DM sources the best cross-time linkage is:
1361 DR <-> 581 CY
Note that the DM articles were written by Ed Greenwood about the wizards Elminster, Mordenkainen, and Dalamar meeting in his home on Earth to discuss the magical affairs of Faerun, Oerth, and Ansalon with reference to various famous wars and other events; and as such provide a strong primary source for cross-time linkages. As such, the dates of these articles can be used to provide an Earth-time linkage as well, should such be desired.
Other cross-time links
Linkages to other settings frequently fall back on the publication date of the earliest release of said setting, combined with in-setting references, or lack thereof, to determine dates. For detailed reasonings, check the link if available - and I have saved a copy in case the link goes down.
Calendars
After all is compared, the following calendars can be synced, with confidence:
- Forgotten Realms: Dale Reckoning (DR), Shou Lung (SL)
- Greyhawk: Olven Calendar (OC), Common Year (CY)
- Dragon Lance: Ansalonian Calendar "Alt Cataclius" (AC)
- Spelljammer: Time of Unity (TU), Anno Vulkarus/Promo Novo (AV), Astromundi Chronos (AstroC)
5043 OC = 581 CY = 358 AC = 1361 DR = 2611 SL = 465 AV = 856 TU = 9/1299 AstroC
This corresponds roughly to the 1990's on our Earth, for those interested.
Best Answer
If you are looking at the timeline from 1st to 5th edition, then it is a timespan of about 132 years.
1st edition was set in 1357 DR, while 5th edition (The Sundering) is placed at 1484 DR, and the Adventurers League play starts in 1489 DR.
This places it in line with the Phandelver, since Hotenow erupted in 1451 DR.
Numbers taken from http://forgottenrealms.wikia.com/wiki/Forgotten_Realms which purports to take the dates from the various box sets and info releases from TSR/WotC.