Proto-Indo-European verbs

Proto-Indo-European verbs reflect a complex system of morphology, more complicated than the substantive, with verbs categorized according to their aspect, using multiple grammatical moods and voices, and being conjugated according to person, number and tense. In addition to finite forms thus formed, non-finite forms such as participles are also extensively used.
The verbal system is clearly represented in Ancient Greek and Vedic Sanskrit, which closely correspond in nearly all aspects of their verbal systems, and are two of the most well-understood of the early daughter languages of Proto-Indo-European.

Basics

The reconstruction of verb conjugation in Proto-Indo-European is controversial. The system described here is known as the "Cowgill–Rix" system, which explains fairly well the data found in most subfamilies of Indo-European. However, this reconstruction encounters significant difficulties when applied to the Anatolian and to some extent the Tocharian branch. For this reason, this system is often thought to have formed not in PIE proper, but at a later stage, after Anatolian and possibly Tocharian had split off. Even so, there is no consensus concerning what the ancestral system of verb conjugation prior to the split-off of Anatolian looked like, and which Anatolian differences are innovations vs. archaisms.
The Cowgill-Rix system involves the interplay of six dimensions with the following variables:

3 numbers	singular, dual, plural
3 persons	first, second, third
2 voices	active, middle
4-5 moods	indicative, subjunctive, optative, imperative, possibly injunctive
3 aspects	imperfective, perfective, stative
2 tenses	present, past

Further, participles can be considered part of the verbal systems although they are not verbs themselves, and as with other PIE nouns, they can be declined across seven or eight cases, for three genders and three numbers.

Building blocks

Roots

The starting point for the morphological analysis of the PIE verb is the root. PIE roots are morphemes with lexical meanings, which usually consist of a single vowel flanked by one or more consonants arranged to very specific rules.

Stems and stem formation

Before the final endings – to denote number, person, etc. – can be applied, additional elements may be added to the root. The resulting component here after any such affixion is the stem, to which the final endings can then be added to obtain the conjugated forms.

Athematic and thematic stems

Verbs, like nominals, made a basic distinction based on whether a short, ablauting vowel -e- or -o-, called the thematic vowel was affixed to the root before the final endings added.
In the case of the thematic conjugations, some of the endings differed depending on whether this vowel was present or absent, but by and large the endings were the same for both types.
The athematic system is much older and exhibits ablaut within the paradigm. In the descendant languages, athematic verbs were often extended with a thematic vowel, likely because of the complications resulting from the consonant clusters formed when the mostly consonant-initial endings were added directly onto the mostly consonant-final stems.
Consequently, the athematic verbs became a non-productive relic class in the later Indo-European languages. In groups such as Germanic and Italic, the athematic verbs had almost gone entirely extinct by the time of written records, while Sanskrit and Ancient Greek preserve them more clearly.

Proposed endings

At least the following sets of endings existed:

Primary endings used for:
* Present tense of the indicative mood of imperfective verbs.
* Subjunctive mood
Secondary endings used for:
* Past tense of the indicative mood of imperfective verbs.
* Indicative mood of perfective verbs.
* Optative mood
Stative endings used for
* Indicative mood of stative verbs.
Imperative endings used for
* Imperative mood of all verbs.

Note that, from a diachronic perspective, the secondary endings were actually the more basic ones, while the primary endings were formed from them by adding a suffix, originally -i in the active voice and -r in the middle voice.
The more central subfamilies of Indo-European have innovated by replacing the middle-voice -r with the -i of the active voice.
Traditional accounts say that the first-person singular primary ending is the only form where athematic verbs used a different ending from thematic verbs. Newer accounts by and are similar, with the proto-forms modernized using laryngeal notation.
Sihler, however, notes that many of the most archaic languages have third-person singular forms missing a t and proposes an alternative t-less thematic ending along with the standard ending. Greek and Balto-Slavic have t-less forms in thematic actives, whereas Vedic and Hittite have t-less athematic middle forms.
uses the t-less forms as the starting point for a radical rethinking of the thematic endings, based primarily on Greek and Lithuanian. These proposals are still controversial, however.

Active eventive endings

Middle eventive endings

Stative endings

A second conjugation has been proposed in Jay Jasanoff's h₂e-conjugation theory. Svensson suggests *-h₂éy for the second and third dual stative endings, on the basis of evidence from Indo-Iranian, Tocharian, and Gaulish.

Verb aspects

The standard reconstruction of Proto-Indo-European verbal morphology outlines a system primarily defined by aspectual opposition rather than tense. However, in the most ancient attested Indo-European languages, this system had largely faded and a new emphasis on a past—non-past distinction had emerged. The most archaic known Indo-European language, Hittite, primarily distinguishes between past and non-past forms and bears no trace of any earlier aspectual opposition. There are other Indo-European languages that have largely simplified their verbal morphologies, though it is unusual for an exceptionally ancient Indo-European language such as Hittite to bear little trace of the presumed importance of aspect within PIE. According to the linguists Jesse Lundquist and Anthony Yates, the evidence from Hittite implies the existence of an earlier system primarily centered around lexical rather than grammatical aspect. Yates and Lundquist note the wide variety of verbal classes in Core Indo-European that appear to lack much semantic difference, implying a merger of earlier formations.
Proto-Indo-European verbs belonged to one of three aspect classes:

Stative verbs depicted a state of being.
Eventive verbs expressed events. These could be further divided between:
* Perfective verbs depicting actions viewed as punctual, an entire process without attention to internal details, completed as a whole or not completed at all. No distinction in tense was made.
* Imperfective verbs depicting durative, ongoing or repeated action, with attention to internal details. This included the time of speaking; separate endings were used for present or future events in contrast to past events.

The terminology around the stative, perfective and imperfective aspects can be confusing. The use of these terms here is based on the reconstructed meanings of the corresponding forms in PIE and the terms used broadly in linguistics to refer to aspects with these meanings.
In traditional PIE terminology, the forms described here as stative, perfective and imperfective are known as the perfect, aorist and present systems:

Stative = Perfect
Perfective = Aorist
Imperfective = Present

The present/imperfective system in turn can be conjugated in two tenses, described here as present and past but traditionally known as present and imperfect. The traditional terms are based on the names of the corresponding forms in Ancient Greek, and are still commonly encountered. Furthermore, there is a separate secondary-verb form commonly known as the "stative" and marked by a suffix *-eh₁-, which has no connection with the stative/perfect described here.
The following table shows the two systems of terminology.

Process	Aspect	Aspect	Tense	Tense
Stative	Stative	Perfect		Perfect tense
Eventive	Perfective, punctual	Aorist		Aorist tense
Eventive	Imperfective, durative	Present	Present	Present tense
Eventive	Imperfective, durative	Present	Past or tenseless	Imperfect tense

In Proto-Indo-European, the aspects had no tense meaning, which later developed in the descendant languages. In Ancient Greek, for example, the perfect carried the meaning of a state resulting from a past action, but the PIE stative referred to the state alone. Likewise, the aorist, though having a tense-like meaning in Ancient Greek, had none in PIE. Perfective and stative verbs were effectively tenseless, or indifferent to time.

Eventive verbs

The perfective and imperfective aspect classes are together known as eventive, or verbs that depict events, to distinguish them from stative. Both shared the same conjugation, with some small differences. The main difference was that imperfective verbs allowed the use of special present-tense endings, while perfective verbs only allowed the default tenseless endings.
The present tense used the primary eventive endings, and was used specifically to refer to present events, although it could also refer to future events. The past tense referred to past events, and used the secondary eventive endings. Perfective verbs always used the secondary endings, but did not necessarily have a past-tense meaning. The secondary endings were, strictly speaking, tenseless, even in imperfective verbs. This meant that past endings could also be used with a present meaning, if it was obvious from context in some way. This use still occurred in Vedic Sanskrit, where in a sequence of verbs only the first might be marked for present tense, while the remainder was unmarked. If the verbs were subjunctive or optative, the mood markings might likewise be only present on the first verb, with the others not marked for mood.
In Ancient Greek, Armenian and Indo-Iranian, the secondary endings came to be accompanied by a prefixing particle known as the augment, reconstructed as *e- or *h₁e-. The function of the augment is unclear, but it was not a fixed part of the inflection as it was in the later languages. In Homeric Greek and Vedic Sanskrit, many imperfect and aorist verbs are still found lacking the augment; its use became mandatory only in later Greek and Sanskrit.
Morphologically, the indicative of perfective verbs was indistinguishable from the past indicative of imperfective verbs, and it is likely that in early stages of PIE, these were the same verb formation. At some point in the history of PIE, the present tense was created by developing the primary endings out of the secondary endings. Not all verbs came to be embellished with these new endings; for semantic reasons, some verbs never had a present tense. These verbs were the perfective verbs, while the ones that did receive a present tense were imperfective.