What is the blockchain scalability problem, and how do sharding and rollups address it?

Each block has limited space, so when demand is high, fees spike and throughput is capped — sharding splits the chain into parallel chains, and rollups bundle many transactions into one layer-1 submission, both increasing capacity.

A block can only hold so many transactions, and nodes only produce blocks so often. When lots of people want in, they compete for the scarce block space by bidding up gas prices, so fees soar — effectively pricing users out and putting a ceiling on how many people the system can serve. That tension between openness and capacity is the scalability problem. Two mainstream remedies:

• Sharding — turn the chain into a "blockchain of blockchains," with a main chain coordinating several parallel chains, so there is far more block space overall and more transactions can be processed at once.
• Rollups — a layer-2 approach that executes many transactions off the base chain and then submits them in a compressed batch to the layer 1, deriving security from that base layer while multiplying effective throughput.

Both aim at the same goal: serve more users without fees rising without limit.

Go deeper:

• Scaling — ethereum.org — on-chain scaling (sharding) versus off-chain approaches (rollups and others).