The following is a list of monolingual dictionaries, both online and offline, that are generally considered to be the standard, authoritative dictionaries of the given language -- either in terms of sheer size and depth (e.g. the OED) or because it's the one that every native speaker turns to.

There are still some languages and some dictionaries missing, so if you have any suggestions, please use the Report function (at the bottom of a post) to notify the moderators.

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DICTIONARIES

Catalan:

DIEC2 - Diccionari de la llengua catalana de l'Institut d'Estudis Catalans - the official, the normative.
GREC - Gran Diccionari de la llengua catalana - non-normative-but-nearly (that is, if a word is not in DIEC but it's here, don't hesitate to use it)

Chinese:

published by the government:
China's (Mandarin)
Taiwan's (Mandarin)
Taiwan's (Taiwanese)

Croatian:

Hrvatski enciklopedijski rječnik (via Croatian Language Portal)

Dutch:

Van Dale (website only gives very limited access to the concise version)

English:

Oxford English Dictionaries:
OED - Oxford English Dictionary (not available for free to non-UK residents; students, however, might have access through university--check your university library's website)
Compact OED
OALD - Oxford Advanced Learner's Dictionary

Cambridge's Advanced Learner's Dictionary
Chambers

the following two (and more) available through dictionary.com:
Random House Unabridged Dictionary
American Heritage® Dictionary

French:

TFL - Trésor de la Langue Française
TLFi: linkable simplified / advanced search enabled
Académie française: 8th edition (complete) / 9th edition (incomplete, A through PIE at the moment)
Larousse
Robert (subscription necessary)

German:

Duden - Deutsches Universalwörterbuch (not available online)
ÖWB - Österreichisches Wörterbuch (not available online)
Kurt Meyer, Schweizer Wörterbuch (not available online)
DWDS - Digitales Wörterbuch der deutschen Sprache
Grimms Deutsches Wörterbuch (out of date but still useful and comprehensive)

Hungarian:
LangBridge Nyelvinformatika - Magyar egynyelvű szótár
Italian:

(Old) De Mauro Paravia
Garzanti (free but registration required)
Hoepli - Gabrielli
Sabatini Coletti

Portuguese:

Dicionário Aulete da língua portuguesa

Romanian:

DEX - Dicţionar explicativ al limbii române

Russian:

Ozhegov - Толковый словарь русского языка

Slovenian:

SSKJ - Slovar slovenskega knjižnega jezika

Spanish:

RAE - Real Academia Español (also available through WordReference by clicking "Definición RAE" after looking up any Spanish word)

Turkish:

TDK (Türk Dil Kurumu [Turkish Language Association])

Ukrainian:

Великий тлумачний словник сучасної української мови