Blockchain Engineer Quoc Le writes down How 2.0 will solve scalibility on Medium:

How does Ethereum 2.0 solve scalability problem?
With current problem and limitations , the question is:

How to create a network that has sufficient number of nodes — while keep the resource(CPU/NET/STORAGE) of each node small enough that all people can be apart of them?
The Ethereum foundation proposes a solution to make nodes can work parallel, it calls “Sharding chain”. It will has multiple shard chains in a system.

What is sharding chain?
Traditionally, there has been a requirement for each node to carry all data on the blockchain. With Sharding they slip blockchain state into number of group , that mean each node only has to carry a small amount of data in order to complete a transaction and each Sharding chain is a separate blockchain that contains separated accounts, state, transactions.

Vitalik Buterin: Ethereum 2.0 (Serenity) With Its Promise of ‘1000X Scalability’ Is ‘Really Not So Far Away’

On Wednesday (31 October 2018), Vitalik Buterin, while delivering the keynote speech at the 4th annual “builders” conference (Devcon4), held this year in Prague, Czech Republic, said that Ethereum 2.0 was “really not so far away.”

Vitalik started by listing what will be in Ethereum 2.0 (“Serenity”), features that Ethereum’s core development team has been “talking about for several years”, “researching for several years”, and “actively building for several years”:

Proof of stake (“Casper”)
Scalability (“sharding”)
Virtual Machine Improvements (EWASM)
Improvements to cross-contract logic
Improvements to protocol economics
The Long Road to Ethereum 2.0 (“Serenity”)
It all began on 15 January 2014 with a post by Vitalik for the Ethereum blog titled “Slasher: A Punitive Proof-of-Stake Algorithm”), in which he described an algorithm called Slasher. Right in the first paragraph, he explained the purpose of Slasher:

“The purpose of this post is not to say that Ethereum will be using Slasher in place of Dagger as its main mining function. Rather, Slasher is a useful construct to have in our war chest in case proof of stake mining becomes substantially more popular or a compelling reason is provided to switch.”

During the summer of 2014, various ideas for improving scalability were explored and eventually abandoned: “Proof of Proof of work”; Hub-and-spoke chains; and Hypercube chains. However, there was still some progress made, such as becoming certain that “algorithms with much stronger properties than existing PoS are possible.”

During 2015 and 2016, while research on Casper and the virtual machine was going on, there were “a series of abortive attempts at solving the core problems.”

Then, in 2017, “The Industrious Age” began. During this period, various strands of research were going on:

“Minimal slashing”
“Conditions (February 2017)”
“Casper FFG (Fall 2017)”
“Casper CBC (2017–2018)”
“Erasure coding data availability ”

What will drive the scaling of the Ethereum network?

Ethereum [ETH] has been struggling to find a footing in price, as the tensions of ICOs selling off their treasury of ETH tokens rise. While many are doing so to fund their products, it is also a known fact that the Ethereum network itself has scalability issues.

According to Ambcrypto, there have been a few proposed solutions to the scaling of the network, which include various features such as sharding and the development of Plasma. This brings the question to the minds of enthusiasts: What will drive the scaling of the Ethereum network?

Primary among the proposed solutions is sharding. Sharding is a concept derived from traditional databases, which breaks the data into pieces that can be stored on different servers.

This, applied to the Ethereum blockchain, would greatly increase scalability by not requiring all nodes that secure the network to be “full nodes”. As transaction and blockchain history is required to be stored across nodes, sharding will effectively allow the security to be maintained without requiring this.

Every node will contain a subset of transactions, and transactions that are not part of the subset will be sourced from other nodes. The trust required in this process will be solved using Ethereum’s “cryptoeconomic incentives”.

Another method for scaling the speed of transactions on the Ethereum network is off-chain transactions. Dependable off-chain transactions can make the Ethereum network as fast as centralized payment processes, as stated by Vitalik Buterin.

However, as the Ethereum network is continually running, it is difficult to test the consequences of these changes on the network without negative results. At the same time, there is constant progress being made on the scalability of the blockchain. It is left to see how these methods will be implemented.