Across:
1 Is (-IS-) following (= positional indicator) polite (CIVIL-) American (-A-) getting into (into = -TION-) trouble (= anagram indicator) at school (-S) (CIVILISATIONS) with the ancient Greeks and Romans (CIVILISATIONS),
10 Wise guys (-MAGI-) in type of philosophy (ism = I-SM) (IMAGISM) and poetry expressed by Joyce, for example (IMAGISM),
11 A (A-) key (music key -F-) hang- (hang = -GANH-) -out (= anagram indicator) for one (-I) (AFGANHI) having roots in Asia (AFGANHI),
12 Used 18 down (used “icing” = ICED) to cover the cake (ICE-) on top of dessert (top of dessert = -D) (ICED),
13 Courteous (CIVIL) to all sides today (CIVIL on all sides of today’s crossword),
15 Exotic berry (ACAI) on summits as cold as ice (on summits/first letters of: as cold as ice = ACAI),
17 By the sound of it, pigeon (COO) on first course (first course = C-) with two types of duck (O+O = -OO) (COO),
19 Something for someone who can’t be patient at night (DAY BED) in the guest quarters (DAY BED),
21 Bitter (ACIDIC) police department (-CID-) consumed by agents (CIA = A-IC) retiring (= reversal indicator) (ACIDIC),
22 Whips up (= reversal indicator) bit of mayo (in Co Mayo = Cong = GNOC-) to go with head of cavolo (head of “cavolo” = -C-) – it’s a salute (-HI) (GNOCCHI) to Italian cuisine (GNOCCHI),
23 No Lent in East Berlin (no “Lent” in “East Berlin” = easbri = SERBIA) or in Central Europe (SERBIA),
25 Drunk (= anagram indicator) lager (lager = A-GLER) around end of afternoon (end of afternoon = -N-) (ANGLER) searching for 27 across (ANGLER is searching for “eel”),
27 Bottled up in feelings (bottled up in “feelings” = EEL) for swimmer (EEL),
29 In part, consider a venture (in part, ‘consider a venture’ = RAVE) that appeals to dancers (RAVE), 30 Reviews (= reversal indicator) English (-E) chapter of history (era = A-RE) about key (music key -G-) (AGREE) match (AGREE),
31 Rip-off (SCAM) cutting of Genus Camellia (cutting of ‘Genus Camellia’ = SCAM),
34 Rejected (= reversal indicator) most (l) evil (“evil” without “l” = evi = -IVE) book (tome = EMOT-) (EMOTIVE) that’s controversial (EMOTIVE),
35 What’s the beef (SIRLOIN) with teacher (SIR-) taking joint (joint /-LOIN of meat) (SIRLOIN),
36 Engineer (CIVIL engineer) built (= anagram indicator) taverns (taverns = SERVANT-) and school (-S) for workers in a state (CIVIL SERVANTS).
Down:
2 Picture (IMAGE-) is very (very) fair (half) (half of “very” = -RY) (IMAGERY) with colourful language (IMAGERY),
3 Flowery type (IRIS) isn’t finished (h), in a manner of speaking (“Irish” without “h” = IRIS),
4 Miami Beach is missing … ahem! (“Miami Beach” is missing… “ahem” = miibah = IAMBIC) A stress for Yeats? (IAMBIC verse),
5 Bill (AD-) is a (-A-) soldier (-GI-) getting around (-O) (ADAGIO) a little slow (ADAGIO),
6 Shakespearean villain (IAGO) produces fragment from Machiavelli – a godsend (fragment from ‘Machiavelli – a godsend’ = IAGO),
7 Silly (= anagram indicator) dunce (dunce = NU-CED) conceals article (-AN-) (NUANCED) that’s carefully worded (NUANCED),
8 Community (CIVIL) guard’s (detective inspector’s = DI’S = DIS-) directive (-ORDER) (CIVIL DISORDER) leads to riot (CIVIL DISORDER),
9 Non-religious (CIVIL) Right (“right” = “rite” = CEREMONY) said (= homophone indicator) (CIVIL CEREMONY) something for those joining union without Father of the Chapel perhaps (CIVIL CEREMONY),
14 Feeds off chefs-d'oeuvre (“feeds” off “chefs-d'oeuvre” = chouvre = VOUCHER) that’s a trade concession (VOUCHER),
16 Don leaves San Diego (“Don” leaves “San Diego” = saieg = AEGIS) for protection (AEGIS),
18 Sickening without Ken's (“sickening” without “Kens” = icing = ICING) topping (ICING),
20 Some mispronounced names (some “mispronounced names” = DNA) of genetic stuff (DNA),
21 Characters covered by Rita Hayworth (characters covered by “Rita Hayworth” = AHA) in Eureka (eureka! = AHA!),
24 Artist (RA-) with instrument (violin) isn't finished (n) (“violin” without “n” = -VIOLI) (RAVIOLI) Italian food (RAVIOLI),
26 One in scrum (one in scrum = LOCK-) at match (bout) loses head (b) (“bout” without “b” = -OUT) (LOCKOUT) in firm dispute that leads to exclusion (LOCKOUT),
27 Serge's (serges = EGRESS) novel (= anagram indicator) is way out (EGRESS),
28 Takeaway (LESS-) opens over roadhouse (opens/first letters of over roadhouse = -OR) (LESSOR) offering rooms for hire (LESSOR),
32 Those in Delhi fiction (those in “Delhi fiction” = HI-FI) includes one of the players (HI-FI),
33 Drift out of fair trade (“drift” out of “fair trade” = arae = AREA) region (AREA).