The Underwood typewriter was essentially the first modern typewriter, defining what most people today think of as an antique typewriter. The Underwood Typewriter was invented by Franz Xavier Wagner. Meanwhile, John Thomas Underwood was a prominent ribbon maker who had significant business with Remington. Remington eventually cancelled the ribbon supply agreement, in lieu of manufacturing their own ribbons.  This led Underwood to purchase Wagner's patents. Thus, in 1895, the Underwood Typewriter Company was born, with headquarters in New York City. So, Remington was partially responsible for creating one of their largest competitors.

Underwood continued to dominate the American typewriter market for many years. In 1927, Underwood merged with the Elliot-Fisher Company, forming the Underwood Elliot-Fisher Company. In 1959, the Italian manufacturer Olivetti bought a controlling interest in Underwood in 1959. They fully merged in October 1963, becoming known in the US as Olivetti-Underwood.

Underwood Nos. 1 & 2
The first two Underwood models never displayed a model number. They can be identified by the frame; the top front corner of the frame curves outward and a ridge runs around the frame at the bottom. The model 1 has 39 keys whereas the model 2 has 42 keys. Models 1 and 2 will display "Wagner Typewriter Company" on the back.

Underwood Nos. 3, 4, & 5
In 1900, Underwood models 3, 4, and 5 replaced models 1 and 2. The Underwood No. 3 is essentially a model 5 with a wide carriage. They were given 12, 14, and 16-inch carriages. In 1903, 18, 20, and 26-inch carriage sizes were introduced under a new serial number scheme. In 1927, an 11-inch carriage was introduced on the model 3, following the first set of serial numbers. The model 4 is a more basic version of the model 5, typing four fewer characters. Models 4 and 5 followed the same serial number scheme. In 1929, beginning with serial number 3000000, all Underwood No. 3 machines followed the serial numbers of the models 4 and 5. Early versions of models 3, 4, and 5 were very similar to models 1 and 2; they were not given a ribbon selector and had the early design on the front plate. Up until late 1905, these machines displayed "Wagner Typewriter Company" on the back. Later versions of models 3, 4, and 5 had the curved pinstriping decals on the front plate as well as a ribbon selector.
Underwood No. 3 #518226
My Underwood No. 3, serial number 518266, has a 12-inch carriage.
Underwood No. 3 #3653785
My Underwood No. 3, serial number 3653785, has a tabulator.
Underwood No. 4 #109012
My Underwood No. 4, serial number 109012, was made in 1906. It is an early machine in the fact that it has the earlier front plate decal design and no ribbon selector. However, it is late enough to no longer make a reference to the "Wagner Typewriter Company" on the back of the machine.
Underwood No. 5 #1938245-5
Underwood No. 5, serial number 1938245-5.

Underwood No. 6, Master, S, & SS
In 1931, the Underwood No. 6 was introduced. It is not labelled as a model 6, but is easily identified by the indented front panel. In October 1937, the model 6 was replaced with the Underwood Master, following the same serial numbers. In 1940, the model S was introduced, followed  by the model SS in May 1946. 
Underwood No. 6 #4227544-11
My Underwood No. 6 with serial number 4227544-11 has an 11-inch carriage.
Underwood No. 6 #4503048-12
My Underwood No. 6, serial number 4503048-12, has a 12-inch carriage.
Underwood Master #M4971302-11
My Underwood Master, serial number M4971302-11, has an 11-inch carriage.

Underwood SX-100 & SX-150
Around 1950, the Underwood SX-100 was introduced. It was also known as the "Rhythm Touch". Advertisements claimed that over 7000000 machines were sold. In October 1953, at serial number 7500000, the SX-150 was introduced (according to Underwood Typewriters by Will Davis, the SX-150 was introduced at serial number 7570000 in 1955). The SX-150 was made until 1957.

My Underwood SX-100, serial number 11-6672421, has an 11-inch carriage.
Underwood SX-150 #7988949
Underwood SX-150, serial number 7988949. This machine displays "Golden Touch", as many Underwood machines made in the 1950s did.

Underwood Touch-Master, Touch-Master II, & Touch-Master Five
The Underwood Touch-Master replaced the SX-150 in 1957 around serial number 8000000. The Touch-Master II soon replaced the first model, and it was subsequently replaced by the Touch-Master Five. These models were produced throughout the early 1960s, most likely ending in October 1963, when Olivetti completely merged with Underwood. I would list more accurate dates and serial number ranges. However, the Typewriter Serial Number Database is entirely incorrect on serial numbers.
Underwood Touch-Master II #11-8388573
My Underwood Touch-Master II, serial number 11-8388573.
Underwood Touch-Master Five #11-9421800
Underwood Touch-Master Five, serial number 11-9421800.

Underwood Standard Portable (Three Bank Keyboard)
In 1919, to compete with Corona portables, Underwood released their first portable typewriter. It had three rows of keys, just like the Corona No. 3. However, there was no need for a folding carriage on Underwood's portable. This three-bank machine was produced until 1929, when over 200000 machines were manufactured
Underwood Standard Portable (Three Bank)
My Underwood Standard Portable (Three Bank Keyboard) is an earlier version. Later machines also have a shift key on the right side of the keyboard. Even earlier machines than this have less of a curve in the typebars resting in the type basket.

Underwood Standard Portable (Four Bank Keyboard), Universal, Junior, F-Model, Champion, Student, Leader, Ace, Correspondent, De Luxe, Finger Flight Champion
In 1926, Underwood released a four-bank portable, beginning a new line of serial numbers and future models. By this time, Remington, Corona, and Royal all had four-bank portables on the market. The coition did not seem to hinder the success of the Underwood machine, though; over 1000000 machines using essentially the same design were produced for over 30 years.
Underwood Standard Portable (Four Bank) #4B97305
My Underwood Standard Portable (Four Bank Keyboard) has the serial number 4B97305. The Underwood Junior is a downgraded, depression-era version of the Four Bank Standard Portable lacking extra features, such as a backspacer. 
A basic Underwood portable released in 1926 is the Underwood Leader. 
Underwood Universal #F1190952
The Underwood Universal is an upgraded Underwood Leader; it introduced touch control and a ribbon selector. My Underwood Universal has the serial number F1190952. The "F" prefix on the serial number is why some of these machines are known as "F-Model" portables.
Underwood Champion
The Underwood Champion is an upgraded Universal in the fact that it has a tabulator. It was introduced in 1934.

In 1949, the Underwood Finger-Flite Champion and Underwood De Luxe were introduced. The De Luxe was the same as the Finger-Flite Champion with paper supports, a metered paper guide, and a variable line spacer. 
Underwood Leader U2409743
In 1949, another Underwood portable body style was introduced. The Underwood Star and Underwood Leader were the basic models, with the Underwood Star having a monotone color scheme. My Underwood Leader has a light brown top cover and dark brown frame. The Underwood Jewell was an upgraded model with a ribbon selector and paper bail. It is also found under the name Hideaway. The Underwood Correspondent had margin sets visible through the paper table. The Underwood Ace combined all features of the models previously listed, and added key-set tabulator. This version was also sold as the Underwood Universal.
Underwood Ace
Here is a slightly different body style implemented on this Underwood Ace.
Underwood Universal
This Underwood Universal has essentially the same body as the Underwood Ace above.
Underwood Leader
Yet another body style can be seen on this Underwood Leader. The Underwood Universal is the upgraded version.