A good question, with a fascinating history. I'll answer from the perspective of electronic materials in materials science. In short, with today's experimental techniques, we can see atoms! In fact, we can even manipulate individual atoms for potentially useful things (more on this later)!

But this was not always the case. One of the earliest records we have on the atom came from Democritus, an ancient Greek philosopher (others like Plato and Aristotle had similar trains of thought). Democritus had a thought experiment. The idea was if you took a material and divided it half, you would have a smaller but identical chunk. If you keep dividing your material, there should eventually be a point where you've reached the smallest representative element of your material. That element is the"atom". In fact "atom" is derived from the Greek word "atomos," which roughly translates to indivisible (it turns out there are even smaller components that make up an atom, but the name stuck; see Standard Model ).

It wasn't until much later in the 18th and 19th centuries that significant progress towards understanding the atom was made. Our understanding of the structure of the atom has vastly changed from the solid-ball model John Dalton first proposed with the development and advancement of quantum mechanics. It is the interplay between theory (e.g., quantum mechanics) and experiment that let's us characterize and engineer materials at the atomic level. Both are part of a feedback loop where e.g., experiment confirms a theoretical prediction or theory explains what is observed experimentally.

Read more about the history and how our understanding of the atom changed here ) (several seminal experiments that contributed to this are described).

The coolest part is the fact that we can see atoms in experiment, and even manipulate them!

Here is the famous image of 35 Xenon atoms precisely positioned to form the letters 'IBM' using a scanning tunneling microscope. Each dot you see is an individual Xenon atom. Read more here and here .

Research on engineering materials at the atomic level is a fast growing area (you might have heard of nanotechnology, for instance). For example, here at UCSB, there is active research on growing materials with atomically sharp interfaces. Here is an example . Each of the balls you see corresponds to a La or Sr atom; the other elements are harder to see because they are smaller/lighter, though you might be able to see the >b>Al and Ti if you squint hard enough). This material was carefully grown using a technique called molecular beam epitaxy (MBE), and is being studied for its rich physics and as the next big electronic material for things like your computer and phone.

Hope this helps!
Best,

Nowadays we can indeed see atoms using advanced technology, like scanning tunneling microscope (STM). This technology allows us to observe, or even move an individual atom. Here is a picture of the silicon atoms that scientists see using STM:
silicon atoms

We can't see them with our eyes because they're too small, but we can build machines that can see them. In particular, x-rays are short enough light waves to be able to reflect off of individual atoms, so using an x-ray camera and shining x-rays through something with an orderly arrangement of atoms (like a crystal), we can see the atoms inside of it by seeing how the x-rays scatter.

Great question. At first, scientists were not sure about what matter looked like on a very small scale. Atoms were just a theory. But scientists made a lot of other theories in chemistry that fit with the idea that matter is made up of atoms. Even though we couldn't prove these theories, many of them were useful. For example, atoms can explain why chemicals always react in the same ratios, since these chemicals are made up of atoms that must group together in the same numbers every time.

By now, scientists have very good evidence that atoms exist. In 1981, the scanneling tunneling microscope (STM) was invented. We can use the STM to see details smaller than one nanometer, including atoms. Wikipedia has some interesting images made using the STM. We even have machines that can arrange atoms in simple patterns--many early examples of this technology show atoms arranged to spell out words or company names, like NIST and IBM.