InuYasha's yōkai blood gives him superhuman strength. No upper limit has yet been shown in the canon, but, according to the Zusetsutaizen Ōgikaiden official character data book, he is at least strong enough to easily lift a boulder with a diameter greater than the height of his body over his head with one hand, and then run many miles with it. Such a boulder would weigh perhaps eight to ten tons. The Ōgikaiden similarly states that he is strong enough to punch through solid steel with his bare hands, as seen in the manga version of the battle with the Noh Mask.
While not as fast as Entei the yōkai-horse that served as Hakudoshi's steed, InuYasha's speed has been shown to easily exceed that of a normal horse; possibly as fast as, if not faster than, a cheetah. Due to the speed he can accumulate in his runs, added with his incredible strength, he is able to cross great distances with a single jump. Altogether, this creates the illusion that he is flying every time he leaps. The Ōgikaiden says he can reach the top of tall cliffs with a single leap as well.
InuYasha's stamina likewise far exceeds human standards, possibly derived from the fact that his yokai blood is that of a dog, dogs naturally having great stamina. He is shown to be able to run tremendous distances at tremendous speeds with minimal effort, and is often seen complaining about how his comparatively weaker human companions always slow him down by needing to rest so much every day.
While his fire-rat robe provides additional protection, a power afforded to anyone wearing it (such as whenever InuYasha gives his robe to Kagome for protection), InuYasha's physique is capable of taking a lot of damage, more than the average human, or even most yōkai. He appears unaffected by severe blunt force trauma, able to withstand a direct blow from a wooden log that broke on contact, and unfazed by the tremendous impact of being thrown hard enough to shatter solid rock. He also possesses tremendous resistance to sharp weapons, able to withstand attacks that slice through trees. While he can be injured by more powerful yōkai and holy weapons, he appears invulnerable to most human ones, able to catch normal arrows out of the air and break steel swords with his bare hands.
Even without his fire-rat robe, he has shown the ability to withstand direct contact with fire and tremendous heat, as well as corrosive substances like acid. When injured, he heals far more rapidly than a human, so much so that attacks powerful enough to severely injure him are rarely fatal, as they would be to humans, if they sustained an equivalent injury. For example, he recovered quickly from numerous serious attacks, including a large fist-sized hole punched through his gut. Unfortunately for InuYasha, his tongue is still susceptible to being burned by curry. He also appears weak against strong smells, as was the case with the ink from the episode "The Cursed Hell Painter".
InuYasha's senses are highly developed, and in some ways, appear to be super-canine rather than superhuman. His senses of hearing and smell are quite likely his most powerful — he is shown to be able to hear what people are whispering from great distances, as was the case when, while perched on a tree, he overheard Akitoki Hojo hiding in a bush and quietly praising Kagome under his breath. His ability to hear is sometimes used for comic relief, as when he overhears Shippo whispering insults about him to Kagome. Likewise, his sense of smell is shown to be extremely sensitive - he can smell hints of blood from miles away and can detect the presence of people by smell long before they can be seen. This is also sometimes used for comical purposes, such as in the episode "The Cursed Ink of the Hell-Painter" where he could be temporarily knocked out by the strong smell of the black ink.
Not much is explicitly said about his other senses. One can however speculate that his eyesight is somewhat enhanced at night, due to the fact that his eyes sometimes are shown to glow, and appear to share some characteristics with cat eyes. The fact that his robes are bright red (one of the colors dogs cannot see) suggest that InuYasha's visual spectrum is at least similar to that of humans. His sense of taste may also be heightened, considering how affected he is by curry.
Besides wielding Tessaiga, which neither Sesshomaru nor most other full-demons can do, InuYasha can perform the following techniques:
Sankon Tessō (散魂鉄爪, Soul-Scattering Iron Claws?, Iron Reaver Soul Stealer)
InuYasha's basic attack, which channels his yoki through his claws, and is powerful enough to tear apart iron, making it easy for him to rip his enemies in half.
Hijin Kessō (飛刃血爪, Hijin Kessō? Blades of Blood)
InuYasha's blood, when combined with the "Soul-Scattering Iron Claws", does damage when thrown. With this attack, he is able to attack his enemies from afar instead of in close combat. InuYasha splashes his own or other's blood on the end of his claws and flings them like multiple shuriken to cleave his foes in two.
This technique basically involves InuYasha jumping down the “Bone Eater’s Well” and traveling back and forth 500 years from the future. How this is done has not been explained and is one of the series most talked about plot-holes. In episode 4 among others, InuYasha traveled to Kagome's time without the aid of a jewel shard. Others besides Kagome and InuYasha have been unable to travel through the well. Specifically, in one episode, Shippo attempts to go through the well but fails. In another episode, Sota (Kagome's younger brother) attempts to go down the well to call InuYasha for help, he also fails.
In the anime he has the ability to float in the sky or ride upon a sparkling dust cloud. Should he choose to do so for an extended period, the pant-legs of his hakama become ephemeral flowing masses-- similar to how one would visualize the wind-driven cloak of a ghost. In the manga, Sesshomaru's legs and the bottoms of his hakama become furry tails that propel him through the air. He also chooses to ride Ah-Un in flight, occasionally in the series.
Due to his lineage, Sesshomaru exhibits the ability of having extremely heightened senses. Sesshomaru has particularly shown his heightened sense of smell, such as being able to tell the Band of Seven were undead due to the scents of soil (graveyard soil in particular), and he was also able to tell that InuYasha's scent changed when the latter temporarily changed into a full demon. Through smell alone, Sesshomaru is also able to learn about events that is transpiring from far away, or even an event that has already happened.  Besides the five basic senses, Sesshomaru's ability in sensing other's auras are exceptional.
Sesshomaru possesses physical strength far beyond that of a normal demon. He was able to easily lift Inuyasha by the throat into the air with one hand in one of their earlier encounters. In addition, Sesshomaru was seen carrying the boulder-sized demon head of Goshinki in one hand with little to no effort. In the InuYasha profiles manga, Rumiko stated that Sesshomaru has Herculean strength, even with only one arm; his strength with only one arm is equal to InuYash's full strength.
His instantaneous power grants him the ability to move faster than the eye can see, both on land and through the air. When he does this in the anime, he leaves behind afterimages. He can also instantaneously close long distances without being detected by others.
The ability to morph into a ball of energy and move himself (and small amounts of other material, including other living beings) over great distances in a short amount of time. This is an ability given only in the anime.
As the son of a daiyōkai, Sesshomaru has inherited similar levels of power and thus is a daiyōkai in his own right; endowed with heightened senses, intelligence, strength, and a myriad of yōkai powers. While he usually appears to be a very handsome human bearing splendid clothing and armour, he can transform into his true form at will. In the anime, he transforms surrounded in a bright red aura or has a tornado circling around him. His eyes then take on a bright red color and yōkai marks on his body widen and become more apparent; the strips on his cheeks which are usually smooth become jagged and change into a somewhat fiercer color. It is similar to InuYasha during his yōkai transformations, however, Sesshomaru transforms much further due to his yōkai blood and becomes a gigantic, crimson eyed,canine with white fur. In the anime he has acid breath as well. It is released in his saliva and spreads like mist that melts anything nearby, and he loses the massive speed he had in his human form.
Being a yōkai, Sesshomaru does not age the way humans do. He has been around for at least a few hundred years. 
Sesshomaru is immune to diseases, poisons, and gases that can paralyze or kill both humans and other normal creatures as well as weaker yōkai. His greatest resistance seems to be toward holy and divine powers; his resistance to purification powers was so strong that he was able to enter Mt. Hakurei, despite its purification barrier (which was the most powerful barrier in the entire series as its reach extended throughout the surrounding area of entire mountain), without being purified and was only slightly weakened to the degree he was unable to perform his signature kenatsu attack in his battles against Jakotsu and Suikotsu. Also when Kagome shot Sesshomaru using one of her Hama no Ya, Sesshomaru caught it using his right hand’s forefingers. Most yōkai would have been purified, but Sesshomaru remained unharmed with not even a scar on either of his fingers. It seems that the only mystical powers that Sesshomaru is not resistant to is yōki such as those of InuYasha’s Kaze no Kizu. Another example of his strong body seems to come from Naraku’s inability to absorb him. As Sesshomaru blasted pieces of Naraku’s flesh, those same pieces wrapped themselves around him. Then Naraku reconnected himself with the pieces of his flesh that covered Sesshomaru and attempted to absorb him to no avail; as when Sesshomaru broke out unmarked from the absorbing process. Most of this happens because Sesshomaru's aura emanates around his body in perfect "completion", due to his demonic purity. InuYasha possesses a far weaker version of such an aura, therefore it cannot repel energy or poisons like Sesshomaru's can, although it does help with his healing abilities. It is also stated by InuYasha that human attacks will not have an effect on Sesshomaru, and thus far the only damage he has sustained throughout the series (not counting when he was in his weakened state within the Hakurei barrier) has been from yōkai.
Despite his overall intelligence, Rumiko Takahashi reveals that his intellect is actually one of his weakest points (despite it being considerably high, it pales in comparison to his other capabilities).