Hybrid warfare
Definition: There is no universally-accepted definition of hybrid warfare; some debate whether the term is useful at all. Some argue that the term is too abstract and only the latest term to refer to irregular methods to counter a conventionally superior force. The abstractness of the term means that it is often used as a catch-all term for all non-linear threats.
Hybrid warfare is warfare which includes some, parts, or all of the following aspects:

A non-standard, complex, and fluid adversary. A hybrid adversary can be state or non-state. For example, in the Israel–Hezbollah War of 2006 and the Syrian Civil War, the main adversaries are non-state entities within the state system. The non-state actors can act as proxies for countries but have independent agendas as well. For example, Iran is a sponsor of Hezbollah, but it was Hezbollah's, not Iran's, agenda that resulted in the kidnapping of Israeli troops that led to the Israel–Hezbollah War. On the other hand, Russian involvement in Ukraine (pre-2022) can be described as a traditional state actor waging a hybrid war (in addition to using a local hybrid proxy) although Russia denies involvement in the 2014 Ukraine conflict.
 Use of combination of conventional and irregular methods. Methods and tactics may include conventional capabilities, irregular tactics, irregular formations, diplomacy, politics, terrorist acts, indiscriminate violence, and criminal activity. A hybrid adversary may also use clandestine actions to avoid attribution or retribution. The methods are used simultaneously across the spectrum of conflict with a unified strategy. A current example is the Islamic State's transnational aspirations, blended tactics, structured formations, and cruel use of terrorism as part of its arsenal.
Flexible and quick response. For example, the Islamic State's response to the US aerial bombing campaign was a quick reduction of the use of checkpoints, of large convoys, and of cellphones. Militants also dispersed among the civilian population. Civilian collateral damage from airstrikes can be used as an effective recruiting tool.
Use of advanced weapons systems and other disruptive technologies. Such weapons can be now bought at bargain prices. Moreover, other novel technologies are being adapted to the battlefield such as cellular networks. In 2006 Hezbollah was armed with high-tech weaponry, such as precision-guided missiles, which nation-states typically use. Hezbollah forces shot down Israeli helicopters, severely damaged a patrol boat with a cruise missile, and destroyed heavily-armored tanks by firing guided missiles from hidden bunkers. It also used aerial drones to gather intelligence, communicated with encrypted cellphones, and watched Israeli troop movements with thermal night-vision equipment.
Use of mass communication for propaganda. The growth of mass communication networks offers powerful propaganda and recruiting tools. The use of fake-news websites to spread false stories is a possible element of hybrid warfare.
Three distinct battlefields. They are the conventional battlefield, the indigenous population of the conflict zone, and the international community.

Other definitions: The Chief of Staff of the US Army defined a hybrid threat as an adversary that incorporates "diverse and dynamic combinations of conventional, irregular, terrorist and criminal capabilities." The US Joint Forces Command defines a hybrid threat as "any adversary that simultaneously and adaptively employs a tailored mix of conventional, irregular, terrorism and criminal means or activities in the operational battle space. Rather than a single entity, a hybrid threat or challenger may be a combination of state and nonstate actors."
The US Army defined a hybrid threat in 2011 as "the diverse and dynamic combination of regular forces, irregular forces, criminal elements, or a combination of these forces and elements all unified to achieve mutually benefiting effects." NATO uses the term to describe "adversaries with the ability to simultaneously employ conventional and non-conventional means adaptively in pursuit of their objectives."
The former US Army Chief George W. Casey Jr. talked of a new type of war that would become increasingly common in the future: "A hybrid of irregular warfare and conventional warfare." According to the 2017-inaugurated European Centre of Excellence for Countering Hybrid Threats, "hybrid threats are methods and activities that are targeted towards vulnerabilities of the opponent" where the "range of methods and activities is wide".

Relation to the grey-zone: The concept of grey-zone conflicts or warfare is distinct from the concept of hybrid warfare, although the two are intimately linked as in the modern era states most often apply unconventional tools and hybrid techniques in the grey-zone. However many of the unconventional tools used by states in the grey-zone such as propaganda campaigns, economic pressure, and the use of non-state entities do not cross over the threshold into formalized state-level aggression.

Effectiveness: Traditional militaries find it hard to respond to hybrid warfare since it is hard to agree on the source of the conflict. An article published in Global Security Review, "What is Hybrid Warfare?" compares the notion of hybrid warfare to the Russian concept of "non-linear" warfare, which it defines as the deployment of "conventional and irregular military forces in conjunction with psychological, economic, political, and cyber assaults." The article partially attributes the difficulty to the "rigid" or static military taxonomy used by NATO to define the very concept of warfare.
To counter a hybrid threat, hard power is often insufficient. Often, the conflict evolves under the radar, and even a "rapid" response turns out to be too late. Overwhelming force is an insufficient deterrent. Many traditional militaries lack the flexibility to shift tactics, priorities, and objectives constantly.

History: When going through the work of philosophers who dealt with propaganda and governance in the last 3,000 years, one can find that hybrid war is not a new concept as many social anthropologists believe today. The combination of conventional and irregular methods is not new and has been used throughout history. A few examples of that type of combat are found in the American Revolutionary War (a combination of George Washington's Continental Army with militia forces) and the Napoleonic Wars (British regulars co-operated with Spanish guerrillas).
One can find examples of hybrid warfare in smaller conflicts during the 19th century. For instance, between 1837 and 1840, Rafael Carrera, a Conservative peasant rebel leader in Guatemala, waged a successful military campaign against the Liberals and the federal government of Central America by using a strategy that combined classical guerrilla tactics with conventional operations. Carrera's hybrid approach to warfare gave him the edge over his numerically-superior and better-armed enemies.
The Soviet Union engaged in an early case of hybrid warfare in 1944. When the Tuvan Army was away in  Europe, fighting along the Red Army against the Third Reich, Moscow annexed the Tuvan People's Republic by pressing the Tuvan government to ask for joining the Soviet Union.

After 1945: The Vietnam War saw hybrid warfare tactics employed by both sides, with the US using the CIA to support civil war parties in Laos and the Cambodian Civil War as well as ethnic groups inside Vietnam for its cause and the Soviet Union supported the Viet Cong militia.

After 1989: The end of the Cold War created a unipolar system with a preponderant American military power. Though that has tempered traditional conflicts, regional conflicts and threats that leverage the weaknesses of conventional military structures are becoming more frequent.
At the same time, the sophistication and the lethality of non-state actors has increased. They are well armed with technologically advanced weapons, which are now available at low prices. Similarly, commercial technologies such as cellphones and digital networks are being adapted to the battlefield. Another new element is the ability of non-state actors to persist within the modern system.

Modern examples: 

2006 Israel–Hezbollah War: One of the most often quoted examples of a hybrid war is the 2006 conflict between Israel and Hezbollah. Hezbollah is a sophisticated non-state actor that is sponsored by Iran. While the group often acts as a proxy for Iran, it has its agenda. It was Hezbollah's policy, rather than Iran's, that led to the kidnapping of Israeli troops, which was the impetus for the war. The war featured about 3,000 Hezbollah fighters embedded in the local population attacked by about 30,000 Israeli regular troops.
The group used decentralized cells composed of guerrillas and regular troops, armed with weaponry that nation-states use, such as anti-tank missiles, rockets, armed unmanned aerial vehicles, and advanced improvised explosive devices. Hezbollah cells downed Israeli helicopters, damaged Merkava IV tanks, communicated with encrypted cell phones, and monitored Israeli troops movements with night vision and thermal imaging devices. Iranian Quds Force operatives acted as mentors and suppliers of advanced systems.
Hezbollah leveraged mass communication immediately distributing battlefield photos and videos dominating the perception battle throughout the conflict. Israel did not lose the war on the battlefield but lost the information battle, as the overwhelming perception  then of Israeli defeat.

2014 ISIL advance into Iraq: The Islamic State of Iraq and the Levant (ISIL) is a non-state actor that uses hybrid tactics against the conventional Iraqi military. ISIL has transitional aspirations and uses irregular and regular tactics and terrorism. In response, Iraq turned to hybrid tactics itself by using non-state and international actors to counter the ISIL advance. The United States was a hybrid participant and used a combination of traditional air power, advisers to Iraqi government troops, Kurdish peshmerga, sectarian militias; it also trained opposition forces within Syria. The hybrid war is a conflict with an interconnected group of state and non-state actors pursuing overlapping goals and a weak local state.