Although I am not expert in this field, but quick literature yielded following results.(This is not elaborate literature search)
latest trends in site specific conjugations methods on proteins

1. Antibody Drug conjugates with drugs and linkers
2. Antibody and target selection
3. Anti-CD30 monoclonal antibody conjugated to the highly cytotoxic drug monomethylauristatin E (MMAE) via modification of native cysteine side chain thiols.[1]
4. Site-specific protein modification via a novel cysteine-containing fusion tag (Cys-tag)[2] or cysteine-engineered antibodies.[3]
5. Modified aptamers[4]
6. Sortase variants[5, 6]
7. Oxime conjugation[7]
8. Microbial Transglutaminase[8]
9. Incorporation of bioorthogonal functionalities[9]
10. Transglutaminase-Catalyzation[10]
11. PEGylation[11]

References

1. Behrens, C.R. and B. Liu, Methods for site-specific drug conjugation to antibodies. mAbs, 2014. 6(1): p. 46-53.
2. Backer, M.V., et al., Cysteine-Containing Fusion Tag for Site-Specific Conjugation of Therapeutic and Imaging Agents to Targeting Proteins, in Peptide-Based Drug Design, L. Otvos, Editor. 2008, Humana Press: Totowa, NJ. p. 275-294.
3. Shinmi, D., et al., One-Step Conjugation Method for Site-Specific Antibody–Drug Conjugates through Reactive Cysteine-Engineered Antibodies. Bioconjugate Chemistry, 2016. 27(5): p. 1324-1331.
4. Wang, R., et al., Using modified aptamers for site specific protein-aptamer conjugations. Chemical Science, 2016. 7(3): p. 2157-2161.
5. Chen, L., et al., Improved variants of SrtA for site-specific conjugation on antibodies and proteins with high efficiency. 2016. 6: p. 31899.
6. Glasgow, J.E., M.L. Salit, and J.R. Cochran, In Vivo Site-Specific Protein Tagging with Diverse Amines Using an Engineered Sortase Variant. Journal of the American Chemical Society, 2016. 138(24): p. 7496-7499.
7. Agten, S.M., P.E. Dawson, and T.M. Hackeng, Oxime conjugation in protein chemistry: from carbonyl incorporation to nucleophilic catalysis. Journal of Peptide Science, 2016. 22(5): p. 271-279.
8. Wakabayashi, R., et al., Protein-Grafted Polymers Prepared Through a Site-Specific Conjugation by Microbial Transglutaminase for an Immunosorbent Assay. Biomacromolecules, 2017. 18(2): p. 422-430.
9. Rashidian, M., J.K. Dozier, and M.D. Distefano, Enzymatic Labeling of Proteins: Techniques and Approaches. Bioconjugate Chemistry, 2013. 24(8): p. 1277-1294.
10. Lin, C.-W. and A.Y. Ting, Transglutaminase-Catalyzed Site-Specific Conjugation of Small-Molecule Probes to Proteins in Vitro and on the Surface of Living Cells. Journal of the American Chemical Society, 2006. 128(14): p. 4542-4543.
11. Nischan, N. and C.P.R. Hackenberger, Site-specific PEGylation of Proteins: Recent Developments. The Journal of Organic Chemistry, 2014. 79(22): p. 10727-10733.

Nice overview. And I agree on entry 1, that I see as THE hot topic. More specific the use of Click Chemistry. Activation by e.g. converting a biomolecule to the coresponding azide. binding dyes and drugs by a covalent bond.

Proteins play myriads of role such as (i) providing structure, strength & stability, (ii) Cell signaling and regulate cellular/physiological activities, (iii)catalyzing reactions (Enzyme), (iv) storage & transport, (v) defend the body from external attack. Modifications of protein are used for studying structure and function relationships for in-vitro & in-vivo research. Protein modifications also find immense potential in therapeutic applications (targeted drug delivery using proteins such as hormones and antibodies that are naturally bear very high specificity for their target) and in diagnosis (Biosensors for cancer detection and maintenance). Protein bioconjugates can be synthesized by using chemical reactions, enzymatic reactions, or genetic engineering technologies. There are several tools for site specific modification of proteins that can be categories into three broad classes such as (i) Chemical modifications of proteins, (ii) Metabolic modifications, (iii) Post translational modifications. The purpose of all these reactions are to selectively install particular modifications on proteins for many biological and therapeutic applications.

The protein modification reactions in current use vary widely in their inherent site selectivity, overall yields and functional group compatibility. The recent trends to overcome the challenges in the existing methodologies for site specific conjugations on proteins will be found out from following most recent works going on across the globe. The antibody drug conjugates (ADCs) is one of the hot topics in protein conjugations

Further Reading
1.      Jenkins N, Modifications of therapeutic proteins: challenges and prospects. Cytotechnology. 2007; 53(1-3) :121-5.
2.      Schumacher D, Helma J, Schneider AFL, Leonhardt H, Hackenberger C. Chemical functionalization strategies and intracellular applications of nanobodies. Angew Chem Int Ed Engl. 2017.
3.      Padayachee ER, Biteghe FAN, Malindi Z, Bauerschlag D, Barth S. Human Antibody Fusion Proteins/Antibody Drug Conjugates in Breast and Ovarian Cancer. TransfusMed Hemother. 2017; 44(5):303-310.
4.      Trads JB, Tørring T, Gothelf KV. Site-Selective Conjugation of Native Proteins with DNA. Acc Chem Res. 2017; 20;50(6):1367-1374
5.      Milczek EM. Commercial Applications for Enzyme-Mediated Protein Conjugation: New Developments in Enzymatic Processes to Deliver Functionalized Proteins on the Commercial Scale. Chem Rev. 2017
6.      Deming TJ. Functional Modification of Thioether Groups in Peptides, Polypeptides, and Proteins. Bioconjug Chem. 2017; 28(3):691-700.
7.      Grant GA. Modification of Cysteine. Curr Protoc Protein Sci. 2017 Feb 2;87:15.1.1-15.1.23.
8.      Drake PM, Rabuka D. Recent Developments in ADC Technology: Preclinical Studies Signal Future Clinical Trends. BioDrugs. 2017; 31(6):521-531.
9.      Zhou Q. Site-Specific Antibody Conjugation for ADC and Beyond. Biomedicines. 2017;5(4).
10.  Xu Y, Jin S, Zhao W, Liu W, Ding D, Zhou J, Chen S. A Versatile Chemo-Enzymatic Conjugation Approach Yields Homogeneous and Highly Potent Antibody-Drug Conjugates. Int J Mol Sci. 2017; 18(11).
11.  Nanna AR, Li X, Walseng E, Pedzisa L, Goydel RS, Hymel D, Burke TR Jr, Roush WR, Rader C. Harnessing a catalytic lysine residue for the one-step preparation of homogeneous antibody-drug conjugates. Nat Commun. 2017; 8(1):1112.
12.  Ohata J, Ball ZT. A Hexa-rhodium Metallopeptide Catalyst for Site-Specific Functionalization of Natural Antibodies. J Am Chem Soc. 2017; 139(36):12617-12622.
13.  Sahota S, Vahdat LT. Sacituzumab govitecan: an antibody-drug conjugate. Expert Opin Biol Ther. 2017; 17(8):1027-1031.
14.  Anami Y, Xiong W, Gui X, Deng M, Zhang CC, Zhang N, An Z, Tsuchikama K. Enzymatic conjugation using branched linkers for constructing homogeneous antibody-drug conjugates with high potency. Org Biomol Chem. 2017; 15(26):5635-5642.
15.  Choudhary S, Barth S, Verma RS. SNAP-Tag Technology: A Promising Tool for Ex Vivo Immunophenotyping. Mol Diagn Ther. 2017; 21(3):315-326.
16.  Chouman K, Woitok M, Mladenov R, Kessler C, Weinhold E, Hanz G, Fischer R, Meinhold-Heerlein I, Bleilevens A, Gresch G, Haugg AM, Zeppernick F, Bauerschlag  D, Maass N, Stickeler E, Kolberg K, Hussain AF. Fine Tuning Antibody Conjugation Methods using SNAP-tag Technology. Anticancer Agents Med Chem. 2017;17(10):1434-1440.
17.  Siegmund V, Piater B, Zakeri B, Eichhorn T, Fischer F, Deutsch C, Becker S, Toleikis L, Hock B, Betz UA, Kolmar H. Spontaneous Isopeptide Bond Formation as a Powerful Tool for Engineering Site-Specific Antibody-Drug Conjugates. Sci Rep. 2016; 6:39291.
18.  Meldal M, Schoffelen S. Recent advances in covalent, site-specific protein immobilization. F1000Res. 2016; 5. pii: F1000 Faculty Rev-2303.
19.  Lim SI, Kwon I. Bioconjugation of therapeutic proteins and enzymes using the expanded set of genetically encoded amino acids. Crit Rev Biotechnol. 2016; 36(5):803-15.
20.  Zheng M, Zheng L, Zhang P, Li J, Zhang Y. Development of biorthogonal reactions and their applications in bioconjugation. Molecules. 2015; 20(2):3190-205.
21.  van Vught R, Pieters RJ, Breukink E. Site-specific functionalization of proteins and their applications to therapeutic antibodies. Comput Struct Biotechnol J. 2014 ;9:e201402001
22.  Rashidian M, Dozier JK, Distefano MD. Enzymatic labeling of proteins: techniques and approaches. Bioconjug Chem. 2013; 24(8):1277-94.
23.  Takaoka Y, Ojida A, Hamachi I. Protein organic chemistry and applications for labeling and engineering in live-cell systems. Angew Chem Int Ed Engl. 2013 ;52(15):4088-106.
24.  Chen X, Henschke L, Wu Q, Muthoosamy K, Neumann B, Weil T. Site-selective azide incorporation into endogenous RNase A via a "chemistry" approach. Org Biomol Chem. 2013 Jan 14;11(2):353-61.
25.  Chen YX, Triola G, Waldmann H. Bioorthogonal chemistry for site-specific labeling and surface immobilization of proteins. Acc Chem Res. 2011; 44(9):762-73.
26.  Hao Z, Hong S, Chen X, Chen PR. Introducing bioorthogonal functionalities into proteins in living cells. Acc Chem Res. 2011; 44(9):742-51. 
27.Stephen C Alley, Nicole M Okeley, Peter D Senter, Antibody–drug conjugates: targeted drug delivery for cancer, In Current Opinion in Chemical Biology, Volume 14, Issue 4, 2010, Pages 529-537.
28.Wing Ho So, Yu Zhang, Wei Kang, Clarence T.T. Wong, Hongyan Sun, Jiang Xia, Site-selective covalent reactions on proteinogenic amino acids, In Current Opinion in Biotechnology, Volume 48, 2017, Pages 220-227.